University of Ghana http://ugspace.ug.edu.gh UNIVERSITY OF GHANA THE EFFECTS OF HOLIDAYS ON THE GHANAIAN EQUITY MARKET BY ALEXANDRA FAFALI KUDJAWU (10341334) THIS THESIS IS SUBMITTED TO THE UNIVERSITY OF GHANA, LEGON IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF MPHIL RISK MANAGEMENT AND INSURANCE DEGREE JULY, 2018 University of Ghana http://ugspace.ug.edu.gh DECLARATION I certify that this thesis is the result of my own research and that; no part of it has been presented for another degree in this University or elsewhere except for those literature, quotations, explanations and summarizations which are duly identified and acknowledged. I bear sole responsibility for any shortcomings. …………………………………….. .………………………………. ALEXANDRA FAFALI KUDJAWU DATE (10341334) i University of Ghana http://ugspace.ug.edu.gh CERTIFICATION We, the undersigned hereby certify that this work was done from candidate’s own research and supervised with the guidelines laid down by University of Ghana for thesis supervision. ………………………………… ……………………………….. DR. CHARLES ANDOH DATE (Principal Supervisor) ………………………………… ……………………………….. DR. SAINT KUTTU DATE (Co-Supervisor) ii University of Ghana http://ugspace.ug.edu.gh DEDICATION This work is dedicated to God who has been my pillar, my dear parents Mr. Alexander Koku Tsidi Kudjawu (of blessed memory) and Mrs. Bridget Mawuena Anthonio-Kudjawu, my siblings Herbert Selikem Kudjawu and Barbara Sika Kudjawu and my entire family especially my favourite grandparents, aunties and uncles. They have all always supported me, been there for me and inspired me to never give up but soar to higher heights. iii University of Ghana http://ugspace.ug.edu.gh ACKNOWLEDGEMENTS I am grateful to my supervisors Dr. Charles Andoh and Dr. Saint Kuttu, who have taught me and guided me in the process of writing this thesis. A special thanks to the staff and faculty of UGBS who taught me a great deal during my postgraduate course. I would like to thank my mom Mrs. Bridget Mawuena Kudjawu and my friends such as Bernard Karikari, Augustine Anokye-Wusu, Kwame Amponsah, Francis Kuditcher and Raphael for their inputs. iv University of Ghana http://ugspace.ug.edu.gh ABSTRACT This thesis sought to determine if the Ghana Stock Exchange (GSE) is a semi-strong efficient market by investigating whether or not the holiday effect exists. This was done by adopting an ARMAX (2, 2) - GARCH (1, 1) model with 𝐺𝐿+ distribution and the results were further used to estimate the VaR and CVaR at 5% significance levels. The data employed were the daily closing prices of the financial main index; GSE-All Shares and GSE-Composite Index (GSE- CI) from the 3rd January, 2007 to 30th December, 2016; a 10-year period. The results from this research show that there is a significant positive pre-holiday effect and a significant positive post-holiday effect which may not be as a result of bearing higher level of risk. These results suggest that investors on the GSE currently trade more on the 7th trading day preceding and on the 7th trading day after a holiday particularly the Farmers day holiday. The associated risk levels for the Farmers day pre-holiday effect was relatively low, however this was not the case for the Farmers day post-holiday effect. The 7th trading day after the Workers day holiday also had a positive and significant returns which was not as a result of risk. The research is important to investors because it will help them to strategize better in order to take advantage of this calendar anomaly discovered on the Ghana Stock Exchange (GSE). Keywords and phrases: ARMAX, calendar anomalies, efficient market hypothesis, GARCH, 𝐺𝐿+ distribution, holiday effect. v University of Ghana http://ugspace.ug.edu.gh TABLE OF CONTENT DECLARATION ................................................................................................................... i CERTIFICATION ................................................................................................................ ii DEDICATION ..................................................................................................................... iii ACKNOWLEDGEMENT .................................................................................................... iv ABSTRACT ......................................................................................................................... v TABLE OF CONTENT ....................................................................................................... vi LIST OF TABLES ............................................................................................................... ix LIST OF FIGURES .............................................................................................................. x LIST OF ABBREVIATIONS............................................................................................... xi CHAPTER ONE ................................................................................................................... 1 INTRODUCTION ................................................................................................................ 1 1.1 Background of the study ......................................................................................... 1 1.2 Problem statement ................................................................................................... 7 1.3 Purpose statement ................................................................................................. 10 1.4 Research objectives ............................................................................................... 11 1.5 Research questions ................................................................................................ 11 1.6 Research hypotheses ............................................................................................. 11 1.7 Significance of the study ....................................................................................... 12 1.8 Research limitations .............................................................................................. 13 1.9 Research delimitations ......................................................................................... 13 1.10 Organization of the study ..................................................................................... 14 CHAPTER TWO ................................................................................................................ 15 LITERATURE REVIEW .................................................................................................... 15 2.1 Introduction .......................................................................................................... 15 2.2 Theoretical review ................................................................................................ 15 2.2.1 Random walk theory ...................................................................................... 15 2.2.2 Efficient Market Hypothesis (EMH) .............................................................. 17 2.2.3 Behavioural finance theory ............................................................................ 19 2.3 Methodological review .......................................................................................... 21 2.3.1 The different methods employed in seasonality .............................................. 21 vi University of Ghana http://ugspace.ug.edu.gh 2.3.2 The different methods employed in holiday effects studies ............................. 22 2.3.3 The method employed in this study ................................................................ 23 2.3.4 𝐺𝐿+ distribution ............................................................................................ 24 2.4 Empirical review ................................................................................................... 25 2.4.1 Calendar anomalies ........................................................................................ 25 2.4.2 Holiday effects ............................................................................................... 26 2.4.3 African stock market ...................................................................................... 31 2.4.4 Brief overview of the Ghana Stock Exchange ................................................ 33 2.4.5 Holidays in Ghana ......................................................................................... 39 2.4.6 Previous researches on the Ghana stock market efficiency ............................. 40 2.4.7 Risk ............................................................................................................... 41 CHAPTER THREE ............................................................................................................. 43 RESEARCH METHODOLOGY AND DESIGN ................................................................ 43 3.1 Introduction .......................................................................................................... 43 3.2 Research design .................................................................................................... 43 3.2.1 Research sample ............................................................................................ 43 3.2.2 Data source .................................................................................................... 44 3.2.3 Data period .................................................................................................... 44 3.2.4 Research data description ............................................................................... 45 3.3 Method of data analysis......................................................................................... 45 3.3.1 Converting closing prices to returns ............................................................... 45 3.3.2 Adjusting returns for thin trading effect.......................................................... 46 3.4 Looking for pattern using event study approach .................................................... 48 3.5 Time series analysis .............................................................................................. 49 3.5.1 Stationarity of time series ............................................................................... 49 3.5.2 ARMA - GARCH models .............................................................................. 51 3.5.3 ARMAX (p, q) – GARCH (x, y) model.......................................................... 52 3.5.4 ARMAX (p, q) model specification as applied in the study ............................ 53 3.5.5 𝐺𝐿+ distribution ........................................................................................... 53 3.5.6 Maximum Likelihood Estimations (MLE) for volatility models ..................... 55 3.5.7 Parameter estimation ...................................................................................... 56 3.5.8 GARCH model specification as applied in the study ...................................... 60 3.5.9 ARMAX-GARCH model testing ................................................................... 61 vii University of Ghana http://ugspace.ug.edu.gh 3.6 Risk metrics .......................................................................................................... 62 3.7 Software used for the estimation of the models...................................................... 63 CHAPTER FOUR ............................................................................................................... 64 EMPIRICAL RESULTS AND ANALYSIS OF FINDINGS ............................................... 64 4.1 Introduction .......................................................................................................... 64 4.2 Time series............................................................................................................ 64 4.3 Descriptive statistics for the adjusted thin trading stock returns ............................. 66 4.4 Residual diagnostic tests ....................................................................................... 68 4.4.1 Autocorrelation test ........................................................................................ 68 4.4.2 Heteroscedasticity test .................................................................................... 69 4.4.3 Stationary test or unit root test ........................................................................ 69 4.4.4 Model specification for ARMAX model ........................................................ 70 4.5 Event study ........................................................................................................... 71 4.6 Specific holiday .................................................................................................... 76 4.7 Risk measures ....................................................................................................... 78 SUMMARY, CONCLUSION AND RECOMMENDATIONS ........................................... 86 5.1 Introduction .......................................................................................................... 86 5.2 Summary .............................................................................................................. 86 5.3 Conclusion ........................................................................................................... 87 5.4 Recommendations ................................................................................................. 89 5.5 Study limitations and direction for further research ............................................... 89 REFERENCES ................................................................................................................... 91 APPENDICES .................................................................................................................. 100 viii University of Ghana http://ugspace.ug.edu.gh LIST OF TABLES Table 4.1: Descriptive statistics of thin trading adjusted stock returns …………………..…67 Table 4.2: Results of Breusch-Godfrey serial correlation LM Test ….…………………..…68 Table 4.3: Results of the test for heteroscedasticity (White test)…...…….……………...….69 Table 4.4: Results of Augmented Dickey-Fuller test with intercept………………………...70 Table 4.5: Model specification for ARMAX (p, q) model ………………………………….71 Table 4.6: ARMAX regression results (pre- and post-holiday effects) ……………….….....72 Table 4.7: ARMAX regression results for specific holidays………………………………...76 Table 4.8: ARMAX regression results for Ghana-specific holidays and non-Ghana specific holidays……………………………………………………………………………………....78 Table 4.9: VaR estimates for thin trading adjusted returns with possible asymmetry in the innovations ………………………………….……....…….……………………...……….....79 Table 4.10: 5% VaR and 5% CVAR estimates for thin trading adjusted returns for the period 03.01.2007 to 30.12.2016 under the various sub-periods…...……………………………….81 Table 4.11: 5% VaR and 5% CVaR estimates for adjusted for thin trading returns for the period 03.01.2007 to 30.12.2016 for the various holiday….…………………...……………82 Table 4.12: 5% VaR and 5% CVaR estimates for Ghana-specific and non-Ghana specific holidays…..……………………………………....…….……………………...……………..84 Table 4.13: 5% VaR and 5% CVaR estimates for pre- and post-holidays of Ghana-specific and Non-Ghana specific holidays…………..……....…….……………………...…………..84 ix University of Ghana http://ugspace.ug.edu.gh LIST OF FIGURES Figure 2.1: Forms of information-based efficiency …………….………………………… 19 Figure 2.2: Annual stock market capitalisation of GSE from 1990-2016……………….…. 36 Figure 2.3: Annual total volumes of equities traded on the GSE from 1990-2016………….. 36 Figure 2.4: Total value of equities traded on the GSE from 1990-2016……………….…… 37 Figure 2.5: Stock market turnover ratio on the GSE from 1990-2011………………………. 38 Figure 2.6: Trend in the number of listed firms on the GSE from 1990- 2013…………………………………………………………………………………. 38 Figure 4.1: Time series plot of daily closing price of GSE index; 2007-2016 …………… 64 Figure 4.2: Time series plot of daily log returns (Jan. 2007- Dec. 2016) ..…………….…. 65 Figure 4.3: Time series plot of daily adjusted for thin trading returns (Jan. 2007- Dec. 2016) ……………………………………………………………………………………………… 66 Figure 4.4: Adjusted for thin trading returns, 5% VaR estimates and 5% VaR exceedances estimates from 2007 to 2016 ...……………………………………………………………. 80 Figure 4.5: Bar chart of 5% of VaR and 5% CVaR estimates of the various holidays……. 83 x University of Ghana http://ugspace.ug.edu.gh LIST OF ABBREVIATIONS AR Autoregressive ARMA Autoregressive Moving Average ARMAX Autoregressive Moving Average Exogeneous CVaR Conditional Value at Risk EMH Efficient Market Hypothesis ES Expected Shortfall GSE Ghana Stock Exchange GARCH Generalized Autoregressive Conditional Heteroscedastic GoG Government of Ghana MA Moving Average MLE Maximum Likelihood Estimation OLS Ordinary Least Squares RSS Residual Sum of Squares 𝑆𝐺𝐿+ Standardized 𝐺𝐿+ VaR Value-at-Risk xi University of Ghana http://ugspace.ug.edu.gh CHAPTER ONE INTRODUCTION 1.1 Background of the study Across the globe today and in every business news carried by all media houses, stock exchange activities are featured prominently. This is corroborated by Kargbo and Adamu (2009), Beck and Levine (2004), Masoud (2013) that the stock market plays a vital role in any country’s economy. Corporations and governments through the existence of stock markets are capable of raising long-term capital to finance expansion of operations and development of new projects (Olweny & Kimani, 2011). With respect to economic importance associated with the development of stock market, investors, academic researchers, regulators and several other stakeholders have and continue to pay keen attention to its operations. This has led to the plethora of works on the stock market aimed at improving its activities so that as often as possible investors can take informed decisions to efficiently maximize their returns while considering relatively lower levels of risk. Available studies relating to the stock market range from the seasonality of the stock market (Yuan & Gupta, 2014; Wasiuzzaman, 2017 & 2018), factors affecting returns, stock market volatility (Andoh, 2010; Kuttu, 2017), efficiency of the stock market (Fama, 1970), the volume of trade (Dodd & Gakhovich, 2011; Pisedtasalasai & Gunasekarage, 2007), to the effect of news or sports on the stock market (Berument & Ceylan, 2012 & 2013), amongst other themes in the stock market literature. According to Esso (2010), an efficient financial system, and a well-integrated banking sector as well as an efficiently developed stock market, provide a better financial service. This is an indication that capital markets are essential because they channel economic resources from savers to the productive sections within the economy. Again, it is important to understand that the economic growth of a nation depends in a large extent on the vibrancy and effective University of Ghana http://ugspace.ug.edu.gh operations of its stock market. This view is supported by Levine (1997) who maintains that a well-developed market is capable of mobilizing domestic savings, boosts higher economic growth and attracts international investors. This goes to say that the evolution of capital market will continue so far as savers continue to search for higher returns and entities such as firms, individuals and governments continue to seek cheaper capital. Hence, the development and efficiency of a stock market is expected to affect economic growth positively in the future (Ntim, Opong, Danbolt & Dewotor, 2011). For this reason, there is certainly no gainsaying that there exists a positive correlation between an efficient stock market and a positive economic growth. Efficiency of the stock market is one of the fundamental concepts in finance that is used to explain and understand the workings of the stock market; it is mainly deployed in explaining the behavioural patterns of stock markets. Efficiency of the market is generally grouped into three categories: allocational efficiency, operational efficiency and informational efficiency (Pilbeam, 2010, p. 125). He further explained that, operational efficiency deals with the situation where trading in securities are carried out instantly, at a low cost and correctly. Allocational efficiency on the other hand, is a mechanism which distributes scarce resources to areas in the economy where they can be most productive. Informational efficiency implies that stock market prices reflect all available information instantaneously. For purposes of this research, an efficient stock market is described as a market that can fully and precisely exhibit all pertinent information in deciding stock prices and if those prices can adjust to new information quickly (Vidanage & Dayaratna-Bandan, 2012). Thus, this concept is based on the availability of information to all markets players. In effect, this is an indication that information plays vital role regarding decision-making in the market. These decisions include: determining prices of financial assets; predicting future movements of financial assets especially stocks; and helping provide signals on trends and changes of those financial assets 2 University of Ghana http://ugspace.ug.edu.gh (Vidanage & Dayaratna-Bandan 2012). This means that one of the conditions of an efficient market is that information should be readily available and its dissemination should also be strong, fast and without any breaks. Fama (1970) provides propositions that help in appreciating capital market efficiencies. The Efficient Market Hypothesis (hereafter, EMH) refers to the notion that capital markets are efficient and that these efficient markets follow the random walk theory, and past information cannot be used to predict the future. Fama (1970) put the EMH into three categories: strong form efficiency, semi-strong form efficiency and weak-form efficiency. The implications of the groupings suggest that no one can beat the market and make abnormal returns no matter the information they have. However, by using past information and publicly available information in conjunction with past information to their advantage, an investor can beat the market in a weak-form efficient market and a semi-strong efficient market respectively. To beat the market in a strong-form efficient market, on the other hand, an investor must be privy to not only all publicly available information and all past information, but also all private information (also called insider information). The Efficient Market Hypothesis (EMH) suggests that an investor cannot make profit by predicting price movement because making predictions are extremely difficult and highly unlikely. That is to say that the arrival of new information is the engine to the change in price levels. An efficient market must therefore adjust its prices quickly and, on the average, without any form of biasness. Ideally, stock market returns should not diverge from its mean or average on special occasions such as holidays, or a specific day(s) of the week or a particular month(s) in a year on a regular basis. This means that there should be no reason why on average, returns in April should be higher than that of December; or returns on Thursday should be higher than on Monday; or average returns just before a holiday should be greater than average returns on 3 University of Ghana http://ugspace.ug.edu.gh other normal trading days. To sum it all up, average daily or monthly returns are expected to be the same all throughout the year. The implication of market efficiency has recently come under attack with the discovery of anomalies. An anomaly by its definition is an occurrence or situation that a prevailing theory cannot be used to explain. Closely related to the stock market, an anomaly could be described as changes in stock market returns that are not in accordance with the EMH. For this research, calendar anomalies as defined by Alagidede and Panagiotidis (2009) as “the tendency of financial asset returns to display systematic patterns at certain times of the day, day, week or month” is adopted. Over the past decades and even beyond, there have been several studies that discovered the existence of several anomalies in the returns on the stock market (McGowan & Ibrihim, 2009; Alagidede & Panagiotidis, 2009; Marshall & Visaltanachoti, 2010; Chen & Chien, 2011; Mensah, Bokpin & Owusu-Gyamfi, 2016). These anomalies are used to debate the EMH. The anomalies which are also referred to as “calendar effects” exist in different forms, for instance, higher average stock market returns in January – the January effect (Fauntas & Segredakis, 2002; Marshall & Visaltanachoti, 2010; Chen & Chien, 2011), higher average returns on Fridays and lower average returns on Monday compared to other weekdays - the weekend effect (Al-Loughani & Chappell, 2001), higher average returns at the ends and beginnings of months - the turn-of-the-month effect (Abd. Majid, 2017), higher average returns in a particular month as compared to the other months in the year – month-of-the-year effect (Alagidede & Panagioditis, 2009), higher average returns before a holiday – the pre-holiday effect (Ariel, 1990, Kim & Park, 1994; Dodd & Gakhovich, 2011; Alagidede, 2013), relatively higher returns on a particular day of the week – day-of-the-week effect (McGowan & Ibrihim, 2009). There are many explanations that have contributed to these calendar anomalies in literature. However, two of such explanations are mood of investors and buying-selling strategies of 4 University of Ghana http://ugspace.ug.edu.gh investors. According to Dodd and Gakhovich (2011), “it is possible that investors get a positive mood before long weekends and holidays which leads to changes in trading patterns and in turn leads to change in returns.” This is corroborated by Gama and Vieira (2013) who mentioned that mood also tends to influence the behaviour of stock market players since the variables that are closely related to happiness have a positive and significant relationship with stock market returns and to the volume of trading activity. Also, Meneu and Pardo (2004) suggest that the pre-holiday effect in terms of the liquidity measure is as a consequence of the unwillingness of small investors to buy on days before a holiday. Even though these researchers and many have used mood and other details to explain the phenomenon of calendar anomalies they still conclude that the exact source is still unclear. The ‘holiday effect’ is one of the most mysterious and baffling of all the seasonal anomalies (Pearce, 1996; Brockman & Michayluk, 1998). Contrary to the discovery of these calendar anomalies by previous studies as illustrated above, Brooks (2008, p. 454) stated that one cannot at first glance conclude that the discovery of these calendar anomalies is an implication of market inefficiencies even though he acknowledged that these phenomena seem to be a contradiction to the EMH. The author further gave two reasons to support the arguments raised; the excess returns recorded on different days of the week, different months of the year and different holidays in a year could be as a result of time- varying stock market risk and when the excess average returns are employed in a trading strategy where the transaction costs have been taken into account, it was likely the excess average returns would not generate net gains. Hence, the presence of the transaction costs and heteroscedasticity could be possible explanation for these documented phenomena; calendar anomalies. The different studies together with their outcomes and explanations associated with the calendar anomalies make their study and examination even more crucial to all stakeholders in 5 University of Ghana http://ugspace.ug.edu.gh the financial industry. This has succinctly been emphasized by Jahfer (2015) and Lim, Ho and Dolley (2009) that the study of calendar anomalies is important to financial managers and investors, as well as, others who have keen interest in developing a trading strategy that will lead to profits eventually. Decision making in investing on the stock market is imperative and must be done with due diligence. A rational financial decision maker in making decisions must not only be mindful of the returns but also the risk of returns. It is important to identify if there exist variations in terms of the risk of returns and to determine if a low or high return is associated with either a low or high risk at a given period. The discovery of (a) risk pattern(s) in returns might serve as a compass to investors in option pricing, portfolio optimization, valuation and the appropriate risk management strategies. The EMH plays a pivotal role in the economy of Ghana especially in the financial market. More importantly it aids in comprehending how the stock market works in Ghana. Consequently, if the concept of predicting an effect of holidays in the stock market returns is successful, this will cast a doubt on the EMH, especially the weak-form and semi-strong. For an emerging market such as that of Ghana, it is important that the fact of whether it is an efficient market or not is established which will provide evidence such that the best or most appropriate investment strategies could be adopted to help both the academic and practitioners community (Mensah et al., 2016). With a growing volume of empirical studies on the discovery of calendar anomalies across the globe, not much attention has been given to the holiday effect, most especially for emerging markets such as Ghana. To better comprehend the patterns of the stock market in Ghana’s stride towards efficiency, it must be formally established whether there is indeed a holiday effect on the Ghana Stock Exchange. Thus, this research seeks to answer questions relating to the nature of patterns of returns and possibly risk around holiday seasons. 6 University of Ghana http://ugspace.ug.edu.gh 1.2 Problem statement The EMH has currently become one of the significant areas in financial literature and as a results, there exist a lot of researches on this concept (Mlambo & Biekpe, 2007; Lee, Lee & Lee 2010; Jovanovic, Andreadakis & Schinckus, 2016, Jackson & Kremer, 2007; Hung, 2009). These are evidences that the EMH has generated an earnest interest in both the investment and academic communities (Mlambo & Biekpe, 2007). One important area that has developed and attracted many of such academicians is the discovery of the calendar anomalies. Calendar anomalies are said to be the likelihood for returns of financial assets to display systematic patterns at certain times of the day, a particular day, a specific month (Alagidede & Panagiotidis, 2009; Brooks, 2008, p. 454). Marret and Worthington (2009) pointed out that the existence of seasonality and calendar anomalies in the stock market returns is one of the consistent theme in the literature on market efficiency. Despite the many different types of calendar anomalies discovered in the past years, the most researched calendar anomalies in literature are generally the day-of-the-week and the month- of-the-year (Alagidede & Panagioditis, 2009; Mensah et al., 2016). Another of such calendar anomalies which has not received much attention as the day-of-the-week and the month-of- the-year is the holiday effect which is defined as “the tendency of stock market returns to exhibit significant higher returns before a holiday in comparison with the other normal trading days (Ariel, 1990; Dodd & Gakhovich, 2011; Yuan & Gupta 2014). Studies into the holiday effect on various stock markets are not limited to the United State of America (Kim & Park, 1994; Brockman & Michayluk, 1998, Mehran, Meisami & Busenbark, 2012) where it was first discovered but also other areas like Saudi Arabia (Wasiuzzaman, 2017; 2018); Australia (Marrett & Worthington, 2009); Portugal (Gama & Vieira, 2013), Africa (Alagidede, 2013) and Central and European markets (Dodd & Gakhovich, 2011) replicating that of Ariel (1990), one of the pioneering scholars in that regard. An extensive review of these 7 University of Ghana http://ugspace.ug.edu.gh studies showed diverse results, others reported similar results as Ariel (1990) which recorded significantly positive effect of holidays on stock returns (Kim & Park, 1994; Brockman & Michayluk, 1998; Marrett & Worthington, 2009; Dodd & Gakhovich, 2011) yet a few observed no effect (Alagidede, 2013). Additionally, with respect to the methodology, various methods have been employed by previous research papers that studied the existence of a holiday effect in various stock markets. For instance, Pettengill (1989), Ariel (1990) and Kim and Park (1994) calculated the mean and variance of the daily returns as well as their respective t-statistics or chi-square to determine if there existed a difference in their average returns. Later studies such as Marrett and Worthington (2009), Alagidede (2013), Dodd and Gakhovich (2011) went a step further to estimate a simple Ordinary Least Square (OLS) dummy regression model to check the significance and equality of means. However, in their study Chien, Lee, and Wang (2002) pointed out that the OLS method might not have been a suitable approach for testing the seasonality in stock markets because of its empirically invalid assumptions. The assumptions that the OLS method requires is that the error term of the stock returns or data must be normally distributed, serially uncorrelated and have a constant variance (homoscedastic). These properties may, however, not hold in reality in terms of financial data. In effect, the use of OLS regression may result into questionable findings (Brooks, 2008, p. 386). According to Wasiuzzaman (2017) and Yuan and Gupta (2014), the ARMA (p, q) - GARCH (p, q) rather appears to be a better model than the OLS regression to test seasonalities since it has the capacity of treating autocorrelation and time-varying variance in the data (heteroscedasticity). The ARMA part of the model takes care of the autocorrelation whilst the GARCH part takes care of the heteroscedasticity. Further, whereas more studies were conducted in developed countries (Lakonishok & Smidt, 1988; Ariel, 1990; Brockman & Michayluk, 1998; Mehran, Meisami & Busenbark, 2012), a 8 University of Ghana http://ugspace.ug.edu.gh few were done in developing countries (Coutts & Sheikh, 2002; Alagidede, 2013). African stock markets, referred to as emerging, has received relatively low attention in regards to holiday effect on the stock market. An example is a study conducted by Alagidede (2013) where he discovered that out of the six African countries, South Africa was the only country that had a pre-holiday effect during the period of study. A variety of studies on the efficiency of the Ghana Stock Exchange (GSE) suggest that the Exchange is fully inefficient despite the various reforms and installation of a more sophisticated information technologies to facilitate trading activities (Frimpong, 2008). The indication of this inefficiency serves as a perfect focal point for breeding of market anomalies. A market can become an efficient market if investors try to beat the market as a result of inefficiencies discovered (Malkiel, 2003; Frimpong & Oteng-Abayie, 2006). A further examination into the Ghanaian financial literature shows that majority of the studies conducted in the field of market efficiency employed the standard efficiency test such as the correlation test, run test, Augmented Dickey-Fuller test, random walk models, GARCH models (Magnuson & Wydick, 2002; Appiah-Kusi & Menya, 2003; Simons & Laryea, 2005; Jefferis & Smith, 2005; Ntim et al., 2011) amongst other tests and models. In the Ghanaian context, however, the existing works on calendar anomalies have concentrated mostly on the month-of-the-year effect and the day-of-the-week effect. The research conducted on Ghana Stock Exchange by Alagidede and Panagiotidis (2009) found that the day-of-the- week effect occurred on Friday and the month-of-the-year effects in April using an OLS regression model. However, the authors documented that both seasonalities disappeared when only recent information was used (employing a rolling framework) during the period of study (1990-2004). However, Mensah, Bokpin and Owusu-Antwi (2016) extended the period of the study used by Alagidede and Panagioditis (2009) to 2012. The authors used 3 methods, sample 𝑡-test and found that a significant day-of-the-week effect existed for all the days except 9 University of Ghana http://ugspace.ug.edu.gh Thursday with Tuesday being the highest. Tuesday still documented the highest and significant return on average when the dummy OLS regression model was adopted. Finally, the authors used GARCH (1, 1) to capture the seasonality in the volatility and found out that the return process was close to non-stationary. Wiredu and Mamuna (2015) investigated the monthly effect on stock returns of Accra Brewery Limited using the Kruskal-Wallis test and the regression on periodic dummies and revealed that there was no evidence of the month-of-the- year seasonality in the stock returns. Moreover, the data set adopted in stock markets where the holiday effect was discovered in other parts of the world, cannot be used to explain the behaviour of investors on the Ghanaian stock market. This is because the variables used in such studies could have been influenced by distinct factors which are peculiar to their individual territories. Ghana, just like any other emerging market is characterized by illiquidity, low number of listed firms as well as thin trading which are unique to her environment and as such needs a unique study to focus on the holiday effect on the GSE. This study focuses on holiday effect while examining the efficiency of the stock market through behavioural approach of investors to discover if there are abnormal returns as a result of a holiday occurring, using an ARMAX (p, q) - GARCH (1, 1) model, an extension of the ARMA (p, q) - GARCH (1, 1) model proposed by Wasiuzzaman (2017) and Yuan and Gupta (2014). 1.3 Purpose statement The motive behind this research is to examine whether the stock market returns in Ghana are affected by holidays and on which particular day before or after the holiday. This will inform stakeholders whether on days prior to or after holidays investors act in a pattern that 10 University of Ghana http://ugspace.ug.edu.gh continuously or persistently affect the average daily stock market returns and to determine which particular holiday affects the stock market returns whilst considering the risk. 1.4 Research objectives This research specifically seeks to accomplish these objectives: i. To investigate the existence of pre-holidays effects on the Ghana Stock Exchange. ii. To investigate the existence of post-holidays effects on the Ghana Stock Exchange. iii. To investigate the specific holiday effects on the stock market returns of the Ghana Stock Exchange. iv. To determine the risk exposure of the holiday effect on the Ghana Stock Exchange. 1.5 Research questions The following questions are to be answered to be able to achieve the objective of this study: i. Are the pre-holiday effects on the stock returns on the Ghana Stock Exchange positive and significant? ii. Are the post-holiday effects on the stock returns on the Ghana Stock Exchange negative and significant? iii. What specific holidays in Ghana exhibit the holiday effect on the Ghana Stock Exchange? iv. Are the holiday effects as a result of bearing higher levels of risk on the Ghana Stock Exchange? 1.6 Research hypotheses The following research hypothesis are also formulated and examined based on the research questions above: i. 𝐻01: There is no significant difference between the average returns of pre-holidays 11 University of Ghana http://ugspace.ug.edu.gh (post-holidays) and the average returns of other normal trading days. 𝐻11: There is a significant difference between the average returns of pre-holidays (post-holidays) and the average returns of other normal trading days. ii. 𝐻02: There is no significant difference amongst the average pre-holidays (post- holiday) returns for each holiday. 𝐻12: There is significant difference amongst the average pre-holidays (post- holidays) for each holiday. 1.7 Significance of the study This research will be of much importance in the following respects and for the following reasons. First and foremost, this study and its outcomes would aid interested stakeholders especially investors and stock traders to better comprehend the activities of the stock market. They would help them to make a strong, better and informed case for business and investment decisions by taking advantage of the anomaly to make profitable tradings if discovered. In addition, this study will help the regulatory body; Security and Exchange Commission (SEC) to determine the presence of inefficiencies in the market. It will equally help for the purposes of the implementation of policies and regulations leading to the control and stabilization of the performance of the stock market in regards to the improvement of efficiency if an anomaly is discovered. This will eventually help policy makers to attract, restore and maintain investor confidence in the Ghana Stock Exchange. Furthermore, this research will inform the government and other regulatory bodies of how holidays affect the stock market. One of the indicators of economic performance of any country is the performance on its stock market. This study will therefore help in analysing the country’s economic performance. In the same vein, it will help government who through the Security 12 University of Ghana http://ugspace.ug.edu.gh and Exchange Commission (SEC) regulates the stock market by monitoring its performance. The stock market serves as a hub for both local and foreign investors, as a result, this study will invariably provide the right information to government not only to create a conducive environment for investment in the stock market but also for taxation purposes. Finally, the study is expected to make significant contributions to knowledge on the Ghana Stock Exchange by serving as a fundamental material for academic discourse on the stock market of Ghana. The results of this research will contribute to literature since to the best of the researcher’s knowledge there is very little literature on the effects of holidays on the stock market in Ghana. This research will serve as a basis for further research on this topic and enable academics to use this study as an exploration into this area and help better understand related topic and questions. The findings of this study will add to knowledge in understanding the efficiency of the stock markets, hence adding to the ever-growing literature on calendar anomalies. 1.8 Research limitations Between 1990 to 2006, tradings on the market were done on three (3) days; Monday, Wednesday and Fridays. Currently the trading on the stock exchange is done on all five working days with the exception of those that fall on holidays and this began in 2007. Hence, the decision of the researcher to employ the period between 2007 to 2016 for this study. 1.9 Research delimitations The above stated limitation will be tackled by focusing on data from the period between 2007 – 2016, a ten (10) -year data span and by using daily returns which would be appropriate for time series estimations. 13 University of Ghana http://ugspace.ug.edu.gh 1.10 Organization of the study The rest of the work is organized as follows. Chapter two focused on the review of relevant literature and discussed both the theoretical and empirical writing focused on the subject matter of the study. It further highlighted the extent of methodology review on the methods used by previous studies to analyse the various seasonalities discovered over the years. Chapter three highlighted on the methodology used for this research. It focused on the statistical and mathematical tools that aided the time series analysis utilized in this study. It also described the data and research design adopted in this study. Chapter four comprised of the analysis of this research where it has to do with the analysis of the data and presented the results obtained from the time series models adopted in the study. Chapter five discussed the summary and conclusions drawn from the study, and provided areas for further studies as well as recommendations for policy makers. 14 University of Ghana http://ugspace.ug.edu.gh CHAPTER TWO LITERATURE REVIEW 2.1 Introduction This chapter presents a detailed discussion on the background knowledge relevant to the studies. It reviews both theoretical and empirical concepts related to the topic underpinning this study. It also gives a brief background of the various calendar anomalies discovered in the past decades on the Ghana Stock Exchange. 2.2 Theoretical review This section presents the related theories and serve as a guide to this research by examining the body of theories that have been considered by previous papers. The theoretical literature review establishes what theories already exist, the relationship between them, and to what degree the existing theories have been investigated. 2.2.1 Random walk theory In 1983, a French mathematician and a stock broker, Jules Regnault attempted to use advanced mathematics to analyse the stock market by publishing a book titled “The study of chance and the philosophy of Exchange”. Regnault’s work influenced another French mathematician, Louis Bachelier to publish a thesis paper titled “Theory of speculation” which is believed to have established the ground rules for the use of both mathematics and statistics to analyse the activities that take place on the various stock markets. This work also influenced many more works such as Paul Cootner’s “The Random Character of Stock Market Prices”, “A Random Walk Down Wall Street” by Burton Malkiel, an economist and finally “Random Walks in Stock Market Prices” by Eugene Fama. It is, however, believed that Louis Bachelier was the first to set down the random walk theory and Burton Malkiel gave it a modern interpretation (Jarrett, 2010). 15 University of Ghana http://ugspace.ug.edu.gh Malkiel (2003) suggested that “a random walk is a term loosely used in the finance literature to characterise a price series where all subsequent price changes represent random departures from previous prices”. This theory goes on to say that a time series consisting of changes in stock price does not depend on its past or historical values, in other words the stock prices can be described as having no memory and therefore no one can predict their future outcomes using their past or historical values (Fama, 1965 & 1970). Closely related, is a statistical hypothesis that states that changes in successive price are independent and can be constructed by stating that there is no correlation between the changes in time 𝑡 (current period) and changes in time 𝑡 − 1 (previous period). The basic explanations that can be given to the random walk theory are that in the first place, if the flow of information is unbiased and secondly if the stock prices reflect the said information instantaneously, then the next price change will only reflect the current day’s news which is not affected by the previous day’s price change. Again, this theory suggests that because of the nature of stock prices to change randomly, it is highly not possible to forecast the prices of stock. Campbell, Lo and MacKinlay (1997) went a step further to propose the presence of three different types of the random walk hypothesis which largely depends on the nature of increment; Random Walk 1 (RW1), Random Walk 2 (RW2) and the Random Walk 3 (RW3). The Random Walk 1 model which happens to be the strongest form of the general random walk theory states that the error terms are independent and identical distribution (i.i.d). Hence, past prices cannot be used to predict future prices because they do not contain information about the future. Random Walk 2 model stipulates that the independent distribution of new information, although random, can change over a period of time. Thus, it is not possible for stock prices to have identically distributed increments over a long duration of time. Random Walk 3 which is the weakest form states that there is no correlation amongst error terms. 16 University of Ghana http://ugspace.ug.edu.gh A commonly used method to test for the random-walk hypothesis is to test and determine the existence of unit root in prices of stocks. There are many tests that have been used and they include Dickey and Fuller unit root test, variance ratio and Phillip-Perron test. The random walk hypothesis implies that an uninformed or better still an “unsophisticated” investor with a diversified portfolio should and will earn a rate of return that is on average similar to that of an “expert”. Hence, a random walk theory follower believes that it is highly impossible to predict the future and all that an investor can do is to accept the hypothesis of “Efficient Market Hypothesis” (Jarrett, 2010). 2.2.2 Efficient Market Hypothesis (EMH) Fama presented the Efficient Market Hypothesis (EMH) in 1970 which has since been considered as one of the medial proposition in financial literature. This hypothesis has had a strong empirical and theoretical evidence supporting it so much so that it seemed so difficult to be debunked. However, with the recent upsurge of counter arguments against the EMH the questions about its efficacy and importance have come under severe attacks. Despite all the controversies surrounding it, the EMH is still an essential concept to modern financial literature especially in terms of capital markets being informationally efficient. Efficiency of the market is generally grouped into three categories: allocational efficiency, operational efficiency and informational efficiency (Pilbeam, 2010, p. 125). Operational efficiency deals with the situation where trading in securities is carried out instantly, at a low cost and correctly. Allocation efficiency, on the other hand, is a mechanism which distributes scarce resources to where they can be most productive. As the name suggests, informational efficiency indicates that stock market prices reflect all available information instantaneously. Fama (1970)’s EMH provides propositions that helps in appreciating capital market efficiencies. The EMH, thus, refers to the assumption that capital markets are efficient and the 17 University of Ghana http://ugspace.ug.edu.gh notion that these efficient markets follow the random walk, and past information cannot be used to predict the future (Fama, 1970). Fama (1970) put the EMH into three forms that are conditional to three types of information: strong form efficiency, semi-strong form efficiency and weak-form efficiency. These groupings imply that no one can “beat the market” and make abnormal returns no matter the information they have. However, by using past information and publicly available information as well as past information to their advantage, an investor can beat the market in a weak-form efficient market and a semi-strong efficient market respectively. On the other hand, in order to beat the market in a strong-form efficient market, an investor must be privy to not only all past information, all publicly available information but also all private information or insider information. The three forms of the EMH are illustrated in Figure 2.1. Out of these 3 forms of EMH, the weak form of the EMH is believed to be the most acceptable due to the attention and weight it has drawn from the academic society (Jarrett, 2010). Fama (1970) used three points to define the weak form which are tests for return predictability, event studies and tests of private (often known as insider) information (Jarrett, 2010). The EMH states that it is extremely difficult and highly impossible to predict stock prices precisely because of the assumption that the market participants are rational and the determination of the stock prices are as a consequences of the changes in demand and supply. It is prudent to note that when stock prices follow a random walk theory or model it does not imply that the stock market with relatively rational investors is efficient (Malkiel, 2003; Dupernex, 2007). 18 University of Ghana http://ugspace.ug.edu.gh Source: Information and the Levels of Market Efficiency (from Sharpe, Alexander & Bailey, 1999, p. 94) Figure 2.1: Forms of information-based market efficiency 2.2.3 Behavioural finance theory In the last century, concepts in the financial literature have undergone a process of rapid evolution from a traditional concept to a more modern one, this is evident especially in the case of the modern portfolio theory which was developed in the mid-1950s and the consecutive financial models that have emerged and played relevant roles in this rapid development. Theories in the traditional finance have often had their basis on the assumptions of efficient market, rationality and profit maximization (Fama, 1970). In the traditional financial literature and as a standard, individuals are expected to make rational decisions. These rational decisions are emphasized through the models and techniques associated with traditional finance. The assumption of rationality has especially attracted the attention of the modern finance theorist since rationality is required to standardize investment preferences. These theories have failed to account for the abnormal and chaotic situations that have arisen in the actual market but 19 University of Ghana http://ugspace.ug.edu.gh rather pay attention to what a theoretical and objective market situation should be. It is for this reason that the assumption of traditional finance has been under attack since its discovery and the issue of whether humans make rational decisions or not has been under investigation. Human beings who are social creatures are unique in their values and also tend to make decisions that are in accordance with their behaviour and emotions – solely on objective factors. According to Huang, Shieh and Kao (2016), decision making by human beings always begins with behavioural finance. Over the past years and not too long ago, behavioural finance has become a widely popular and relevant concept in financial literature. Despite the recent attention placed on this concept by the academic community, it would not be wrong to say that behavioural finance has always existed since time immemorial; in the subconscious minds of humans through their consumption and investments related activities and engagements. Behavioural finance has its roots firmly in the fields of psychology, economics, finance and sociology (Shindler, 2007; Huang, Shieh & Kao, 2016). It tries to intertwine all these different fields into a single field and recommend a new direction for traditional finance theorists. One of the attempts that behavioural finance has tried to make is to better comprehend how investors behave and explain how investor behaviour affects stock market returns (Dodd & Gakhovich, 2011). Shefrin and Statman (2012) further explained that the field of psychology include emotions, aspirations, culture, cognition and perceptions of fairness. Behavioural finance argues that behaviours and mood are among the many other factors that affect humans in the shaping of their investment preferences. There is no doubt that market participants have been exhibiting “irrational” attitudes as a matter of fact in the past and this is arguably supported by Malkiel (2003) who is of the view that mistakes are bound to be made as a result of collective judgment of investors. This leads to the occurrence of pricing irregularities and prediction over a time period and their persistence lasts for a short period. Again Malkiel (2003), argues that the 20 University of Ghana http://ugspace.ug.edu.gh existence of a holiday effect is a violation of both the semi-strong and strong form of efficiency because of the patterns of returns around holidays. As a result, an investor either adopting the technical approach or the fundamental approach can earn abnormal returns and this implies that in an efficient market no such anomaly should exist. 2.3 Methodological review This section begins with a detailed discussion on the various and different methods that previous studies on calendar anomalies adopted. It continues with methods that were strictly adopted in holiday effects studies and ends with a discussion on the method (ARMAX- GARCH) which was used in this study. 2.3.1 The different methods employed in seasonality Alagidede (2013) opined that the various methods employed by previous studies on stock market anomalies can generally be categorized into 4 groups and are stated below: In the first group, studies such as French (1980), Gibbons and Hess (1981) and Ariel (1990) calculated the returns, means and variances for each time period (be it daily, weekly, monthly or yearly), estimated a simple OLS dummy regression and examined the significance and equality of the mean returns using standard t, F tests or ANOVA. This class of studies did not pay attention to the properties of time series and therefore the results could not be reliable due to the misspecification effects and data generation process (Alagidede, 2013). The second group of studies based on the OLS dummy regression model estimated the mean returns, conducted a hypothesis test by using t-statistics and chi-squared calculated with the use of heteroscedasticity-consistent standard errors and ignore the distributional properties of the data employed. The next group test the normality of the returns and if the returns were found to be normally distributed, a 𝑡-test and 𝐹-test or an ANOVA test or a non-parametric tests were used to test the equality of the means returns. Finally, the fourth 21 University of Ghana http://ugspace.ug.edu.gh group used a GARCH model to study the presence of anomalies if the descriptive statistics of the distributional properties are indicating properties of financial time series (leptokurtic, non-normal, asymmetric). 2.3.2 The different methods employed in holiday effects studies Earlier studies associated with the holiday effect on returns on the stock market generally, calculated the returns, arithmetic mean and variance of the daily returns as well as their respective 𝑡-statistics or chi-square to determine the difference of the means. Since to test the holiday anomaly basically involves comparing the returns for the pre-holiday and post- holiday, these classes of studies used a test of equality of mean across the series and amongst the various holidays. This was to determine whether they were significantly different from each other. Depending on the nature of the data and the number of groups under which the entire dataset had been categorized, the appropriate equality of mean test was adopted. This group of studies inadvertently, did not necessarily rely on the normal distribution assumptions of the data on returns. Later studies went a step further to estimate a simple OLS dummy regression model to check the significance and equality of mean returns. This method is referred to as conventional method by Alagidede and Panagioditis (2009) and essentially entails testing a null hypothesis which states that all returns for each period (normal days, pre-holiday days and post-holiday days) are all the same whereas the alternate hypothesis which is a rejection of the null hypothesis states that one of the sub-periods is statistically different (either positive or negative) from the other two sub-periods. The daily returns are computed and the whole sample under consideration is divided into normal trading days, days preceding a holiday (pre-holiday days) and days after a holiday (post-holiday days) with the use of dummy variables representing each period. This last group of studies made assumptions on the 22 University of Ghana http://ugspace.ug.edu.gh normality of the data however, it ignored the time varying heteroscedasticity problem present in the data which is as a result of volatility clustering and leptokurtic. 2.3.3 The method employed in this study In the opinion of Brooks (2008, p. 386), the assumptions of the existence of homoscedasticity and serial correlation associated with the OLS linear models do not usually hold in situations involving financial time series. Financial time series is immanently non-linear in nature and therefore, linear statistical models such as the OLS regression model cannot be used for predictions and statistical inferences (Mohapatra, Majhi & Sapathy, 2017). Unlike the other economic time series, financial time series exhibit certain unique characteristics which are currently regarded as “stylized facts” such as leptokurtic, leverage effects and volatility clustering (Brooks, 2008, p. 380). These stylized facts have therefore, led to the use of more appropriate and sophisticated non-linear models. Hence, due to the limitations and challenges faced by the OLS linear regression model its estimated parameters may infer wrong conclusions for financial time series data (Yuan & Gupta, 2014; Chien, Lee & Wang, 2002). Yuan and Gupta (2014) suggest the ARMA (p, q)-GARCH (x, y) model as a more appropriate model to model the holiday effect. They opined that the ARMA part of the model cater for the serial correlation whereas, the GARCH models the volatility clustering and leverage effects observed mostly in financial time series. Greene (2003, p. 619) and Makridakis and Hibon (2000) support this view by saying that the ARMA models are found to be relatively the best in modelling especially macro-economic and financial time series. Therefore, the family of ARMA models are efficient for forecasting time series. One of such families is the Autoregressive Moving Average Exogenous (ARMAX) model, which is an ARMA model with other time series inputs included as exogenous variables. GARCH models have in the past decades become important and have played a vital role in financial time series data particularly 23 University of Ghana http://ugspace.ug.edu.gh when it has to do with the analysis and forecasting of volatility (Angabini & Wasiuzzaman, 2011). 2.3.4 𝑮𝑳+ distribution According to Andoh (2009), researchers impose the normal distribution on such models and he suggested that other heavy-tailed distributions could be considered as alternatives. Such heavy-tailed distributions include Student’s 𝑡, Normal Inverse Gaussian (NIG), 𝐺𝐿+ distribution and 𝐺𝐿− distribution. The tail of a distribution is very crucial in financial studies and therefore GARCH models which assumes normality often fail to model and describe accurately the leptokurtic form of such distribution (which is one of the properties that is usually observed in series consisting of returns) (Andoh, 2009). Hence, in dealing with most financial returns it is not wrong to say that the normal distribution also referred to as the Gaussian distribution is not a good fit for its analysis (Nidhin & Chandran, 2013). In their work, Andoh et al. (2018) showed that the 𝐺𝐿+ regression based on the 𝐺𝐿+ distribution had a higher predictive power as compared to the following other models: 𝐺𝐿−, Probit and Logit models. They showed that for all the 1000 paths generated and in terms of the following goodness of fit measurements: the 𝐺𝐿+ model had the least average mean square error and the least expected Akaike Information Criteria and had the second least expected Bayesian Information Criteria. They argued that the Probit and Logit models usually assume that the binary (multinomial) outcomes follow an elliptical distribution which implies that these models tend to impose a certain symmetry and shape without acknowledging the actual data structure. Unlike the above mentioned models, the 𝐺𝐿+ distribution allows for the data structure (independent variables) to determine its own shape and skewness. 24 University of Ghana http://ugspace.ug.edu.gh Andoh (2010) used these distributions: normal, 𝐺𝐺+, 𝐺𝐺+, Student- 𝑡, Normal Inverse Gaussian (NIG), 𝐺𝐿+, 𝐺𝐿− to estimate the Value at Risk (VaR) for some German stocks and found out that the 𝐺𝐿+ and the 𝐺𝐿− gave the best estimate for a properly chosen skewness parameter. Since return series are asymmetric and heavy tailed this study assumes that the returns have a 𝐺𝐿+ distribution so that the data will be allowed to choose its own skewness and shape that is in line with the data that was used for the study. 2.4 Empirical review The section introduces a general analysis of literature on calendar anomalies and holiday effects. It continues with a discussion on the general overview of stock markets in Africa and more specifically associated literature on the stock exchange in Ghana. It ends with a section on risk associated with the loss of returns. 2.4.1 Calendar anomalies Despite many empirical evidence that the stock markets are efficient and rational, many studies have documented the existence of calendar anomalies (Fields, 1931; Dodd & Gakhovich, 2011; Alagidede, 2009; Mensah et al., 2016). This discovery has for the past three decades remained an area of increased interest for researchers since its existence have been evidently discovered in most of developed capital markets in the world. According to Shahid and Akbar (2009), calendar effects or anomalies are the anomalies of stock prices that are attributed to the calendar effects. A study by Fields (1931) is considered as the first documentation of the existence of seasonalities. He analysed the weekend effect and showed that Saturdays had the tendency to record higher returns than on Fridays and Mondays. Subsequent to the study carried out by Fields (1931), many more anomalies have been discovered (Dodd & Gakhovich, 2011) and are used to test the efficiency of various stock markets. Some of such anomalies are the day-of- the-week (Alagidede & Panagioditis, 2009; Mensah et al., 2016), month-of-the-year 25 University of Ghana http://ugspace.ug.edu.gh (Alagidede & Panagioditis, 2009), holiday effect (Dodd & Gakhovich, 2011; Marret & Worthington, 2008; Yuan & Gupta, 2014; Wassiuzaman, 2017 & 2018). Dodd and Gakhovich (2011) observed that in the studied markets, some of the anomalies have either disappeared totally or have weakened over time. He explained that this phenomenon could be as a result of the various publications and academic researches on these anomalies. He further illustrated that “awareness of an anomaly after a publication increases the number of investors which in turns weakens the anomaly”. Therefore, when a lot more investors trade on an anomaly to attain abnormal profit, the anomaly weakens or disappears. This claim is supported by Philpot and Peterson (2011) who explained that particularly the day-of-the-week effect had gradually disappeared since 2003 and attributed it to the fact that investors had incorporated these patterns in their trading strategies with the widespread of its knowledge of existence. Arisss, Rezvanian and Mehdian (2011) found a Friday-type effect and a statistically significant December effect in the Gulf Cooperation Council (GCC) stock markets when they examined calendar anomalies. The most researched calendar anomalies include the day-of-the week and the month-of-the-year (Alagidede & Panagioditis, 2009; Yuan & Gupta, 2014; Mensah et al., 2016). 2.4.2 Holiday effects Apart from the day-of-the-week and the month-of-the-year effects, another well-documented calendar anomaly in the stock market is the holiday effect (Yuan & Gupta, 2014). Generally, holiday effects in stock markets are said to occur when the returns on a day or on few days (differ from studies to studies and could be a day or 5 days) before a holiday exhibit a pattern that are usually abnormally higher than the returns on other regular trading days. Yuan and Gupta (2014) defined “the holiday effect as empirical evidence that stock returns on the day preceding a holiday tend to be abnormally higher than those of other trading days”. According to Pearce (1996) and Brockman et al. (1998), “the ‘holiday effects’ are one of the most 26 University of Ghana http://ugspace.ug.edu.gh mysterious and baffling of all the seasonal anomalies”. The holiday effect is defined as the strong tendency for equities to experience abnormally large returns just prior to holidays (Brockman et al., 1998; Ariel, 1990). A holiday is defined by Lakonishok and Smidt (1988) as a day when trading would normally have occurred but did not as a result of the holiday occurring. Studies on the holiday effect on the various stock market returns have been approached from various perspectives; in terms of specific holidays such as Ramadan effect, Halloween effect, Chinese Lunar New Year effect, religious and secular holidays effects, firm size effect, and industry level effects among others. These variations of holiday effect as employed in various studies are discussed in the remaining paragraphs of this section. Early studies (Ariel, 1990; Pettengill, 1987; Lakonishok & Smidt, 1988; Kim & Park, 1994) show that holiday effect exists in developed countries and other studies including Alagidede (2013) indicate the presence of holiday effects in developing markets. For example, Yuan and Gupta (2014) examined the Chinese Lunar New Year (CLNY) holiday effect in some major Asian stock markets: Japan, South Korea, Taiwan, Hong Kong, China as well as India for a period of 1st September, 1990 to 28th March, 2012. The authors used an ARMA (1,1) - GARCH (1,1) model to investigate the daily stock index returns for each market and concluded that in all the Asian stock markets a positively significant pre-Chinese Lunar New Year effect is observed. They also employed the ARMA-GARCH-in-mean (ARMA-GARCH-M) model to examine if the abnormal returns observed before the CLNY holiday was as a result of a reward for risk. From their findings they observed that whereas the higher returns in other markets are caused by unknown factors as well as conditional risk, the higher returns in China were as a result of compensation for high risks levels. They argued that previous studies that ignored the distributional properties of the returns series and adopted the OLS dummy regression model did not acknowledge the reality of this property. 27 University of Ghana http://ugspace.ug.edu.gh Alagidede (2013), on the other hand, investigated the presence of pre-holiday effects in six African countries and the implication on stock market efficiency by using OLS dummy regression model. By estimating a regression model and examining the significance of the mean and variance of the returns series, South Africa was the only country that showed significantly high pre-holiday returns. Alagidede (2013) opined that the discovery of a pre-holiday effect within the period of study could have been as a result of the closing effect which is usually characterized by high returns for observed financial assets at market closing and good mood usually exhibited by investors around holidays. Contrary to previous studies such as Tonchev and Kim (2004) on the holiday effect in Central and Eastern European countries, Dodd and Gakhovich (2011) documented abnormal positive and significant post-holiday returns as well as the usual pre-holiday effect. Their paper applied OLS regression and found out that there was no single industry that was responsible for this effect however, the Christmas and New Year holidays were the most common holidays which produced the highest and significant returns. They finally concluded that the diminishing trend of the pre-holiday effect observed was an indication of the improvement in the level of market efficiency of the countries considered over the period of the study. Kim and Park (1994) provided further evidence that there was pre-holiday effect in all the three indices; AMEX, NSE and NASDAQ in the US stock market, in the UK and the Japanese stock markets. They set out the trading days before, after and all other trading days as a study group and compared their daily mean returns and used the 𝑡-statistics to test the equality of mean between the 3 groups. The pre-holiday effect discovered in the US for the period 1963-1986 was not as a result of the January Effect because the New Year holiday was deliberately omitted. The authors showed that the holiday effect was not only prevalent in the US where it was first discovered but also in other countries (UK and Japan) and used a dummy regression model to prove that the results in UK and Japan markets where independent of the U.S. market. 28 University of Ghana http://ugspace.ug.edu.gh Ariel (1990) used daily stock index returns from the Centre for Research in Security Prices (CRSP) from two indices: value weighted index and equally weighted index for the period of 1963-1982 to investigate only the pre-holiday effect in the US. He calculated the means and variances for the two stock indices alongside their 𝑡-statistics and found out that the average returns for the pre-holiday were about nine to fourteen times higher than the mean of the normal trading days for equally and value weighted indices respectively and were statistically significant. Ariel (1990) compared the variances for the two groups of the two indices and concluded that despite the constant and high pre-holiday returns documented, they were not as a result of bearing extra levels of risk. Lakonishok and Smidt (1988) documented that much higher and more significant pre-holiday returns than those of normal trading days existed in the Dow Jones Industrial Averages (DJIA) of the New York Stock Exchange (NYSE). Mehran et al. (2012) looked at how Jewish holidays affected the stock market returns in the United States of America even though there were a relatively small number of individuals who observed Jewish holidays. The researchers found that during the period from 1990 to 2009 there was evidence of abnormal returns that were 32 times greater on 9 Jewish holidays than the other normal trading days of the year. They argued that the market participants act differently depending on the holiday that is being observed as a result of the mood associated with the holidays and discovered that joyous holidays had higher and positive returns whilst returns were insignificant and negative on solemn holidays. Marret and Worthington (2009) investigated the Australian Stock Market and concluded that the pre-holiday effect existed and they were on average five times higher than that of the regular trading days. They investigated the holiday effects in the various industry levels and concluded that the retail industry had the highest and significant average returns and could not state 29 University of Ghana http://ugspace.ug.edu.gh categorically if the anomaly discovered was as a result of the mood of investors as iterated by other studies such as Mehran et al. (2012). Wasiuzzaman (2018) also performed a similar study as Yuan and Gupta (2014) where the study sought to find the relationship between Hajj pilgrimage on the Tadawull All-Shares Index (TASI) and other industrial indices of the Saudi stock market. She used ARMA (1,1) - GARCH (1,1) model from January, 2010 to August, 2014 and found that the Hajj period had a significant increase in volatility for all the indices except for that of the agricultural, petrochemical, food and retail sectors and an insignificant and negative impact on the mean return of all the sector indices and the TASI. However, Wasiuzzaman (2017) had established the fact that TASI of Saudi stock market exhibited a Hajj effect. Over the past years, various explanations have been attributed to the existence of holiday effects. Firstly, the existent relationship between the holiday anomaly and other calendar anomalies. This is to say that holiday effect occurs as a result of other calendar anomalies such as the day-of-the-week effect or the month-of-the-year which had earlier been discovered. Researchers such as Lakonishok and Smidt (1988), Ariel (1990) and Liano, Marchand and Huang (1992) attempted to use this explanation and concluded that the high returns observed on days preceding a holiday were not as a result of the occurrence and existence of the other calendar anomalies. Again, holidays affect the mood, demeanour, attitude, and daily experiences of persons who observe them (Mehran, Meisami & Busenbark, 2012). It is believed that the euphoria which accompanies holidays affects the mood and demeanour of most investors as such they would act in a way to affect the activities of the stock market. The euphoria that accompanies holidays is believed to eventually lead to short covering and a general and impulse buying pressure (Jacobs & Levy, 1988; Thaler, 1987; Lahav, Shavit & Benzion, 2016). Wright and Bower (1992) are of the view that judgements of investors are likely to originate from their moods, whence a bad mood and a good mood could lead to 30 University of Ghana http://ugspace.ug.edu.gh pessimism and optimism respectively. Therefore, emotions and moods associated with the various holidays are believed to have the tendency to exert influence on decisions of investors and eventually their stock market attitudes. However, according to Keim (1989), Pettengill (1989) and Lahav, Shavit and Benzion (2016), the holiday effect discovered over the years was neither as a result of euphoria nor short-sellers as suggested by previous researchers but could be as a result of an effect of just the market closing and they termed the “closing effect”. Overall, the holiday effect can be put into two forms; pre-holidays and post-holidays. The pre- holidays are days preceding holidays and post-holidays are days after holidays. However, the pre-holiday effect occurs when the returns of pre-holidays are significantly different from the other regular trading days and the post-holiday effect occurs when the returns of post-holidays are also significantly different from the other trading days. According to Ariel (1990) and Dodd and Gakhovich (2011), they believe that generally before a holiday investors tend to close their short selling positions before a holiday and reopen them after the holiday. This phenomenon tends to increase the pre-holiday returns and decrease post-holiday returns which leads to significant positive and significant negative returns respectively Finally, the holiday effect generally, has been investigated across many stock markets in the world and the results vary from one stock market to another and this is attributed to the variations and differences in the stock markets. The days on the calendar are classified into 3 categories: as pre-holiday, post-holiday, or regular/ normal with no regard to the actual day of the week where generally the returns before holiday (pre-holiday) are positive and significant whereas those of the post-holidays are significant and negative. 2.4.3 African stock market The African stock markets have seen a significant growth in terms of both size and numberover the past years. For instance, the number of stock markets in Africa that are active have appreciated from 5 in the 1990’s to 18 in 2002; there were a total number of 29 formal stock 31 University of Ghana http://ugspace.ug.edu.gh markets in Africa as at 2012 and also propositions to set up and open new ones in many other African countries (UNDP, 2003; Ntim et al., 2011). Currently, the body has 32 members consisting of 27 full members, 3 Associate members and 2 observers. This substantial increase in the number of stock markets in Africa is apparently as a result of the extensive reforms in the financial sector that was undertaken in these areas (Ntim et al., 2011). ASEA, the regulatory body has as its purpose to help with the progress and rise of the African continent with a sound economic growth. ASEA is a non-profit making organization which was established in 1993 in Kenya. The main purpose for its establishment is to foster a unique cooperation among stock exchanges in Africa and to enable its members become significant and key drivers of societal and economic transformation in the continent by the year 2025 (ASEA, 2016 Annual Report). They achieve their purpose by sharing, dissemination and delivering information amongst its members. ASEA has since its establishment put in a lot of efforts and is gradually being recognised as the mouthpiece of African Stock Exchanges. Recently, ASEA launched an African index with the FTSE which serves as an attractive benchmark for investors investing on African stock markets and also as a reference which has helped in the creation of the market vectors Africa (ETF) index (ASEA, 2016). ASEA’s relentless efforts will definitely help consolidate and further improve the fortunes of Stock Exchanges in Africa. Even though it is an established fact that there exists a positive relationship between stock markets development and economic growth (Schumpeter, 1911; Masoud, 2013), African stock markets remain relatively different from their counterparts in the developed markets with the exception of the South African stock market and some stock markets in the Northern part of Africa (Alagidede, 2009; Ntim et al., 2011). Considering all the stock markets in Africa, most of them can be described as emerging or frontier equity markets which is attributed to illiquidity nature of the markets, issues related to thin trading and small capitalizations (Ntim et al., 2011). 32 University of Ghana http://ugspace.ug.edu.gh Despite the many attempts to improve these markets so that they can be at par with their counterparts in the developed markets, most of these African stock markets have the following characteristics; social issues (political instability and bureaucracy), lack of information delivery, lack of good accounting standards, high transactions costs, archaic electronic trading systems which have directly affected real time availability of market information (Ntim et al., 2011). 2.4.4 Brief overview of the Ghana Stock Exchange There had been many attempts by the authorities in Ghana since 1968 to establish a stock exchange, but it was only after the Stock Market Act of 1971 was implemented that the Accra Stock Market Limited was established in 1971. Even though it was a great concept it was only a plan on paper and was never implemented. This failure could be attributed to circumstances such as political instability, unfavourable macroeconomic environment and absence of support from the government. Despite these obstacles, corporate bodies were actively trading through two brokerage firms; National Trust Holding Company Limited and Merban Stockbrokers Limited. The trading was done over-the counter in shares of foreign owned companies. In the 1980s Ghana had to undergo some major structural reforms to rectify and redress the anomalies that the Ghanaian economy faced due to the financial crisis the country underwent from 1983 to 1988. This recovery programme which tackled all aspects of the economy was done under the close watch of the World Bank and International Monetary Fund (IMF). The aim for these reforms especially in the financial sector were first and foremost to liberalize and expand access to long-term capital for investment. A comprehensive Financial Sector Adjustment Programme (FINSAP) was subsequently launched in 1988 and one of its principal objectives was to restructure the financial sector and establish new institutions whose purpose was to revitalize the sector. 33 University of Ghana http://ugspace.ug.edu.gh The stock market was seen as an important part of the economy and thus a great importance was attached to its establishment in Ghana. As such in 1989 the Ghana Stock Exchange (hereafter referred to as GSE) was instituted after a report which contained its feasibility and recommendation was commissioned. The following year, November 1990, the GSE commenced operations with three brokerage firms, 11 listed companies and one government bond (Frimpong, 2008; Mensah et al., 2016). The number of listed firms on the exchange later increased to 13 in 1991, 19 in 1995 and currently they are approximately 43. In 1994, the GSE was adjudged the best index performing stock market amongst the league of emerging markets with 124.3 percent gain in its index level. This performance was an improvement from what was recorded in 1993, where it was the 6th best performing index in emerging stock market with 116 percent growth in capital appreciation. In the year 1995, however, due to high interest rate and high inflation rate, growth of the index declined and recorded a growth of 6.3 percent. Presently, the GSE-CI recorded a negative return of 15.53% and 11.80% in 2016 and 2015 respectively and a total market capitalization of approximately 52,690.99 million Ghana cedis (SEC, 2016 Annual Report). It is believed that the increasing trend of the market’s capitalization over the past years is as a result of the massive increase in the number of listed equities on the exchange. Currently, the main index is the GSE Composite Index (GSE-CI); which is a capitalization-weighted index with a base value of 1000. However, the exchange publishes two indices; the GSE Financial Stock Index (GSE-FI; a market capitalization weighted index of only stocks in the financial sector) and the GSE Composite index (GSE-CI) (a market capitalization weighted index of all common stocks with the exception of those listed on other markets). The ordinary shares (common stocks), preferred stocks or shares and exchange traded funds are the instruments used to trade. Trading on the GSE is done all weekdays except on days that are declared as official holidays and it occurs between the hours of 10:00 a.m. to 15:00 GMT. 34 University of Ghana http://ugspace.ug.edu.gh In 2008, the GSE introduced the automated trading platform known as the GSE Automated Trading System (GATS) to enable the dissemination of information efficiently instantaneously and also through the depository to ensure efficient and quick supply and clearance of traded securities on the Exchange. There are talks by major stakeholders such as the Bank of Ghana, Ministry of Finance, GSE as well as the SEC to develop the commodity market at a subsequent date. This change is aimed at making the stock exchange more effective, efficient and relevant. The GSE has two categories of listing, namely, the main board and the Ghana Alternative Market (GAX). The GAX, since its inception in 2015 is operated as a parallel market and it is aimed at small and medium-sized enterprises (S.M.E.) with strong growth potential (retrieved on 2/1/2018 from www.gse.com.gh ; https://afx.kwayisi.org/gsegh/). The GSE is the major stock market in Ghana and regulated by the Securities and Exchange Commission (SEC) and seeks to build resilient markets for a developing economy (GSE, 2016). The Exchange is also a member of both the World Federation of Exchanges and the West African Capital Market Integration (WACMIC). The GSE equities market capitalisation has increased generally from GH¢ 3.05 million in 1990 to approximately GH¢ 62.91 billion in 2016. It can be seen that in the early years of the existence of the GSE, that is in the 1990s the exchange was relatively small. The GSE did very well in 2004 and this is attributed to the merger of Ashanti Goldfields with AngloGold (GSE market report, 2005). Again, there was a tremendous increase in 2000 from GH¢365.5 million to about GH¢62,917.19 million in 2016. Despite these increments, total market capitalisation reduced by 18.12% from 2014 to 2015. A summarized report of the annual market capitalisation from 1990 to 2016 is shown in Figure 2.2. 35 University of Ghana http://ugspace.ug.edu.gh 70,000 60,000 50,000 40,000 30,000 20,000 10,000 - 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 Year Source: GSE, Annual Report 2018 Figure 2.2: Annual stock market capitalisation of GSE from 1990-2016. Total volume traded depicts the aggregate number of shares traded at a specific period of time and is also used to measure liquidity in an index. The total volume of equities traded annually have over the years not been encouraging attributable to the rising and falling trends recorded. The highest volume traded was recorded in 2008 which stood as at GH¢531.6 million. The summarized report on the total volume traded since 1990 to 2016 is depicted in Figure 2.3. 600,000 500,000 400,000 300,000 200,000 100,000 - Year Source: GSE, Annual Report 2018 Figure 2.3: Annual total volumes of equities traded on the GSE from 1990-2016 36 Volume ('000) Mkt Cap (GH Cedis in millions) 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 University of Ghana http://ugspace.ug.edu.gh Closely related to the volume traded is the total value traded. It is also a liquidity measurement. It is calculated as a product of the volume of the index and its closing price. A summary of the total value of equities traded is shown in Figure 2.4. It is observed that Figure 2.3 is similar to Figure 2.4. Thus, the observations discussed above still hold for that of the total value of equities traded. 500 450 400 350 300 250 200 150 100 50 0 Year Source: GSE, Annual Report 2018 Figure 2.4: Total value of equities traded on the GSE from 1990-2016 The turnover ratio is computed by dividing the total value of shares traded by the market capitalisation for the period under consideration. The highest turnover ratio was recorded in 1998 and the lowest in 2006. This is an indication that GSE’s efficiency increases and decreases through the years. A summary of the trend of turnover ratios is illustrated in Figure 2.5. 37 Total value (GH Cedis in miilions) 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 University of Ghana http://ugspace.ug.edu.gh 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 - Year Source: Researchers’ own calculation, 2018 Figure 2.5: Stock market turnover ratio on the GSE from 1990-2011 Figure 2.6 shows that the trend of the number of companies listed on the GSE has been increasing steadily over the years since 1990. As at 2013, the number of listed firms had almost doubled. 35 30 25 20 15 10 5 0 Year Source: WDI Figure 2.6: Trend in the number of firms listed on the GSE from 1990-2013 38 Number of listed firms Turnover ratio 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 University of Ghana http://ugspace.ug.edu.gh 2.4.5 Holidays in Ghana For the purposes of accomplishing the goals of this research, a holiday is defined as “a day on which the stock market is closed as a result of a public holiday (Marret et al., 2009). Ghana’s annual calendar is characterized by many holidays. These holidays vary and can be grouped into religious, national and international categories. According to the Government of Ghana (GoG), as at 2017, there were about 12 holidays in the 2017 annual calendar. These holidays can also be put into the same three categories stated above: Religious, National and International. The various categories and their related holidays are explained below: Under the religious holidays there are Christmas, Easter, Eid-ul-Fitr and Eid-al-Adha. The Christmas and Easter holidays are generally celebrated by the Christian fraternity to mark the birth and death of Jesus Christ respectively. Christmas, celebrated on the 25th of December of every year is followed by another holiday called Boxing day which falls on the 26th of December. The mood generally associated with these holidays are joyous and also lots of fun and party. Unlike the Christmas holidays, Easter has no fixed date and it falls between the months of March and April. The holidays during Easter are Good Friday and Easter Monday. The Muslims, on the other hand, celebrate Eid-ul-Fitr and Eid-al-Adha whilst the latter is determined by lunar eclipse, the former is celebrated to mark the end of their Ramadan fast. The national holidays in Ghana include Independence day, Republic day, National Founder’s day and Farmers day. Independence day which falls on the 6th of March and marks the independence Ghana won from her colonial masters; Britain. Republic day is celebrated on the 1st of July to commemorate the day Ghana became a republican state. Another national holiday is the birth date of Dr. Kwame Nkrumah, the first president of Ghana which falls on the 21st September and is called Founder’s Day, Farmer’s day which was earmarked to celebrate the works of farmers in the country is celebrated on every first Friday of the month of December. The international holidays include African Union (AU) day which falls on the 25th May, New 39 University of Ghana http://ugspace.ug.edu.gh Year day (1st January) and International Worker’s day (also referred to as Labour day or Workers day is celebrated on the 1st of May). The AU day is set aside to remember the establishment of the African Union (formerly Organization of African Unity), Workers’ day is observed to celebrate the achievements of workers and New Year’s day to welcome the new year (Source: http://www.ghana.gov.gh/index.php/news/52-map-of-natural-resources/306- statutory-public-holidays-2015). It must be noted that these holidays were all not instituted in the same year. They were either adopted or scrapped out when the government of the day deemed them important and necessary. The Founder’s day holiday which celebrates the birth of Kwame Nkrumah; the first president of Ghana, was introduced by the government of the day in 2010. (Source: https://www.modernghana.com/news/419363/1/founders-day-to-be-placed-on-ghanas- holiday-calend.html). One of such holidays scrapped from the calendar of Ghana is the June 4th holiday in 2000 (Source: http://www.ghana.gov.gh/index.php/media-center/features/1950- is-founder-s-day-worth-celebrating). In Ghana, the usual practice is that anytime any of these holidays falls on a weekend (either a Saturday or a Sunday) the holiday is automatically pushed to the next Monday. Hence, holidays that occurred during the weekend will have Tuesday’s returns as their post-holiday returns in this study. 2.4.6 Previous researches on the Ghana stock market efficiency Some researchers regard the Ghana Stock Exchange as weak-form inefficient (Magnusson & Wydick, 2002; Appiah-Kusi & Menya, 2003; Jefferis & Smith, 2005; Alagidede & Pangioditis, 2009; Ntim et al., 2011). In the opinion of Alagidede et al. (2009) in 2005 the GSE All shares Index was disappointingly low and the most critical hurdle for the GSE as at that time was to remove existing impediments to institutional development such as the provision of a wider form of information dissemination and the implementation of a robust electronic trading system. In 40 University of Ghana http://ugspace.ug.edu.gh light of the fact that the GSE played a vibrant role in raising domestic and international capital for economic development, there were reforms which include an efficient and broader dissemination of information through the operation of an electronic and automated trading system in 2008 (Frimpong, 2008). To the best of the researcher’s knowledge there has been few literature (Frimpong, 2008; Alagidede, 2013) on the efficiency of the stock market after these reforms and this research seeks to test the efficiency of the market using the calendar anomalies specifically the holiday effect. This will eventually help investors take advantage of abnormal returns during the various holiday seasons. 2.4.7 Risk Risk has no single definition and as such there exists a number of definitions attributed to the word. A variety of concept of “risk” result from these various fields: economists, behavioural scientists, risk theorists, statisticians and actuaries. Risk, according to Promislow (2012, p. 3) is the possibility that something bad happens but most especially occurrences that will result in financial losses. However, Redja (2014, p. 2) defined risk as uncertainty concerning the occurrence of a loss. Risk can thus be a possibility, a probability or an uncertainty. Even though most often it is associated to a negative theme it can also take the form a positive one (Dorfman & Cather, 2012). Again, Redja (2014, p. 2) states that most corporate risk managers use the term “loss exposure” for the terminology risk because ‘risk’ has different meanings and is ambiguous and he goes ahead to define “loss exposure” as “any situation or circumstance in which a loss is possible, regardless of whether the loss occurs”. The stock market, just like any other financial market is often faced with uncertainties. In this case uncertainty has to do with any negative event that threatens the interest of investors, which is generating more returns from their investment portfolio. Generally, in stock market activities investors are induced and attracted by higher expected returns to take on higher risks, thus they base their decision on a risk-return balance. The risk of any financial asset or investment 41 University of Ghana http://ugspace.ug.edu.gh portfolio is measured by the standard deviation or volatility (Brooks, 2008, p. 383). The expansion of the stock markets across the globe over the years have emphasized the pivotal role that risk management plays and this could be attributed to the fact that risk can never be avoided. Prices of financial assets such as bonds, stocks, commodities as well as derivatives and economic measurements such as interest rate and exchange rate are all susceptible to constant volatility. Consequently, these observed variability leads to significant volatility of their returns over different periods of time which renders forecasting difficult. Volatility is one important variable in financial markets and it aids financial regulators, investors, investment managers and speculators in decision making. Some examples of risks that have been discovered in financial literature include operational risk, credit risk, liquidity risk, political risk, interest rate risk, foreign exchange risk, market risk, hazard risk, strategic risk (IRM, 2002). These are generally categorized into financial and non-financial risk; systematic and non-systematic risk. Since investors do not only consider expected returns but also volatility or risk of returns in their decision making, it is essential that investors on the Ghana Stock Exchange know whether there exists any variations or patterns in the risk of stock returns by the holidays observed over the periods in the past. It is also imperative to know if any abnormal returns discovered is as a result of corresponding high risk taking behaviours. If a particular pattern in the risk is discovered, then it could be easier for such investors to make better investment decisions on the basis of both risk and returns. 42 University of Ghana http://ugspace.ug.edu.gh CHAPTER THREE RESEARCH METHODOLOGY AND DESIGN 3.1 Introduction This chapter consists of research methodology and data sample used in the study. The section presents in details the data that is used for this study and approaches used to model seasonality and the motivation for their usage. This study employed both linear and non-linear models. A detailed discourse on the extension of the conditional heteroscedastic models adopted in this study is provided. The OLS dummy variable regression and ARMAX (p, q) – GARCH (p, q) models with a 𝐺𝐿+ innovation are also explained. 3.2 Research design The study uses a quantitative research design. Boateng (2016, p.135) described quantitative research as “a research that seeks to determine the existence of a relationship between aspects of a phenomenon by quantifying the variation”. This research design was adopted with the aim to identify the existence of a holiday anomaly in the Ghana Stock Exchange. One of the types of quantitative research is the descriptive research design where one tries to establish solely the relationships between variables. In this study, the descriptive research was used with the sole purpose to describe the characteristics of the various variables in the study. 3.2.1 Research sample To achieve the purpose of this study, it utilized mainly secondary data from the Ghana Stock Exchange (hereafter referred to as GSE) which primarily consists of daily closing prices of both the Ghana Stock Exchange All Shares Index and the Ghana Stock Exchange Composite Index (hereafter referred to as GSE-CI). The Composite Index is a volume weighted index of the average closing prices of all the listed equities on the Exchange. The GSE-CI can also be described as a capitalization index with a base value of 1000. Simply put, the amount an equity 43 University of Ghana http://ugspace.ug.edu.gh contributes to the index depends on the market capitalization of the listed company. The GSE All Shares was changed to the GSE-CI on 4th January, 2011. Therefore, at the beginning of this study, the All-Share Index was used and was followed by the GSE Composite index for the period understudied. 3.2.2 Data source The daily stock prices were retrieved from the Ghana Stock Exchange. The Ghana Stock Exchange is the only Exchange in Ghana licensed by the Security and Exchange Commission. Currently, the total number of stocks listed on the GSE is approximately 43 as well as about 21 Licensed Dealing Members. As at 2016, the total market capitalization of the GSE was GHS 52,690.99 million in absolute terms (Annual Report 2016, SEC). The dataset for this study had a total of 2476 number of daily observations. The data used to carry out this empirical study was divided into 3 groups: pre-holiday days (105 observations), post-holiday days (105 observations) and other normal trading days (2266 observations). The researcher believes that in using the daily observations, the best possible results would be determined. This is because the stock prices, as a result of volatility clustering, may exhibit high volatility within short time periods. When identifying the existence of a holiday anomaly within the dataset used, the long period of historical daily data will help capture the various trends both within short and long periods. Furthermore, the researcher believes that the daily trading days currently adopted by the GSE will not change as this model is generally adopted and accepted internationally. 3.2.3 Data period The period of this study is from 3rd January, 2007 to 30th December, 2016; a 10-year period. The year 2007 was used because in 2006 the trading days changed from 3 days to 5 days while the year 2016 was the last full year that data could be obtained. This study used a more recent set of data for its analysis because changes and trends that have occurred in the past would have been documented. The same dataset was also used to examine the risk pattern around holidays. 44 University of Ghana http://ugspace.ug.edu.gh 3.2.4 Research data description The holidays to be considered are as follows: “New Year’s day, Independence day, Good Friday, Easter Monday, May day, African Union day, Republic day, Eid-ul-Fitr, Eid-al-Adha, National Founder’s day, Farmers day, Christmas day and Boxing day”. These holidays are public holidays as defined by the Holidays Act-2001 (Act 601), approved by the country’s Ministry of Interior. However, there is no trading when any of the above falls on a week day due to the observance of the holiday. The dates of the holidays were collected from the internet (Ministry of Interior website as well as time and date website). However, holidays such as “Good Friday” and “Easter Monday” are put together and referred to as Easter holiday whilst “Christmas day” and “Boxing day” are also considered Christmas Holiday. The pre-holidays are described as one or more number of days before a holiday. The post- holidays are also described as one or more number of days after a holiday. Normal trading days are any other trading around which no holiday occurs. However, trading on the Exchange takes place on all days except for days on which a holiday is observed. Hence, trading takes place on normal days, pre-holiday days and post-holiday days. 3.3 Method of data analysis 3.3.1 Converting closing prices to returns Gujarati and Porter (2009, p. 22) opined that most time series data based empirical studies assumed that the underlying time series datasets were stationary. To ensure stationarity of the time series, the log returns of the prices was used for this study. The natural log of the relative price was calculated for each day and a time series made up of continuously compounded returns was generated. A continuously compounded returns Rt time series (Brooks, 2008, p.7), is defined as inter-daily difference of the natural logarithm of the daily prices of the assets (𝑃𝑡) and is illustrated below: 45 University of Ghana http://ugspace.ug.edu.gh  P  R  log t *100 (3.1) t  P   t 1  where; Rt is the continuous compounded return on day 𝑡 ; Pt1 is the closing market price in period 𝑡 − 1 (previous period); Pt is the closing market price in period 𝑡 (current period) and 𝑙𝑜𝑔 is the natural logarithm. The continuously compounded returns are preferred to discrete returns primarily because of the ease of calculation and can easily be compared, since they are given by the first order difference of the logarithmic prices and a time-additivity property, which is useful if we assume a normal distribution of the logarithmic returns (Brooks, 2008, p. 8). The study did not include dividends in the calculation of returns because the prices of the various indices have been adjusted for dividends already. 3.3.2 Adjusting returns for thin trading effect Thin trading is said to occur when stocks do not trade at every consecutive interval (Alkhazali, 2008). Emerging markets are on the whole described as having low liquidity, considerable high volatility, thin trading and perhaps investors that are less informed and have access not only to unreliable but also delayed information (Bekhaert, Erb, Harvey & Vishanta, 1998; Alkhazali, 2011; Yuan & Gupta, 2014). Hence, in testing the efficiency of the stock markets in these emerging markets, considering thin trading effects is imperative because it is usually considered as one of the major characteristics of such markets (Alkhazali, 2011). Most African equity markets have empirically documented pervasive thin trading (Appiah-Kusi & Menyah, 2003; Mlambo & Biekpe 2007; Kuttu, 2017) as such, the continuously compounded returns that are calculated for this study in equation 3.1 were adjusted for the thin trading effect. 46 University of Ghana http://ugspace.ug.edu.gh A lot of studies have shown that infrequent buying and selling of stocks lead to statistical biases in the time series of stock prices (Miller, Muthuswamy & Whaley, 1994; Alkhazali, 2008 & 2011; Loc, Lanjouw & Lensink, 2010). The bias that occurs as a result of the thinly traded stocks are as a result of prices documented at the end of the period of time and these prices have the tendency to represent an outcome of transaction occurring prior to the period in question (Alkhazali, 2008 & 2011; Loc, Lanjouw & Lensink, 2010). Furthermore, the issue associated with infrequently traded stocks is the lack of a price change between two time periods and may be regarded as being caused by the unavailability of a price reaction to a new information (Loc, Lanjouw & Lensink, 2010). In other words, thin trading leads to problems of serial correlation which could affect estimates of the OLS (Pathirawasam & Idirisinghe, 2011). Serial correlation occurs if the error terms are correlated with each other or one another (Brooks, 2008, p. 139). Mensah et al. (2016) proved that GSE index return exhibited a significant autocorrelation up to 20 lags hence it can be concluded that GSE has problems of serial or auto-correlation which could be as a result of the presence of thin trading. Autocorrelation can be said to be “correlation between the members of the series of the observation ordered in time or space (time series or cross-sectional data)” (Gujarati & Porter, 2009, p. 413). A study by Alkhazali (2008 & 2011) and Kuttu (2017) showed that in order to correct for the effect of thin trading, a Moving Average (MA) model that reflects the number of non- trading days is required and subsequently used to calculate the adjusted returns for the effect of non-trading days. The MA is obtained by constructing an Autoregressive model of order 1 (AR (1)) because of the difficulty associated with the identification of the non-trading days in question. Miller, Muthuswamy and Whaley (1994) proposed and showed that the MA model is equivalent to an AR model of order (1), from which the non-trading adjustment can be obtained and found out that thin trading adjustment actually decreases the negative 47 University of Ghana http://ugspace.ug.edu.gh correlation amongst returns (Alkhazali, 2011). Thus, this study follows the methodological approach used by Kuttu (2017) and a method propounded by Miller et al. (1994) to adjust the continuous compounded returns in equation (3.1) for thin trading effects. The following autoregressive model of order 1 (AR (1)) is used: 𝑅𝑡 = 𝛼 + 𝛽𝑅𝑡−1 + 𝜀𝑡 (3.2) where 𝛼 is the constant term, 𝑅𝑡−1 is lag of the returns of order 1 or the previous term of the returns, 𝛽 is the parameter for the past term of the returns, and 𝜀𝑡 is the error term. The adjusted returns for thin trading are calculated by: 𝑎𝑑𝑗 𝜀 𝑅 = ?̂?𝑡 (3.3) 1− ?̂? adj The calculated Rt is the adjusted for thin trading return at time 𝑡 and it is hereafter represented as ?̃?𝑡. These returns ?̃?𝑡 are then classified into pre-holiday returns, post-holiday returns and other normal trading days’ returns. The average daily returns for 𝑛 number of day preceding the holiday and after the holiday are determined. 3.4 Looking for pattern using event study approach Event studies, according to Sharpe et al. (1999) are undertaking to see how returns react to an event or the release of an information. This approach in the end is attempting to see if the returns are high or low, react rapidly or slowly or are just normal prior to the event. The event date for this study was defined as a date on which the holiday was declared and was observed during the period of study. Event studies essentially, are usually employed to investigate the magnitude and relevance of a particular event on another event. The event window was extended to 8 days before and after the event date, in this case the date of the 48 University of Ghana http://ugspace.ug.edu.gh holiday. That is, the event window for this study had 17 days (-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8). Most researchers found the holiday effect on a day before and a day after a holiday (Ariel, 1990; Dodd & Gakhovich, 2011; Lakonishok & Smidt, 1988). But, there could be cases that these investors made preparations long before and in anticipation of the holiday or reacted oppositely after a holiday. Therefore, this enabled the researcher to explore the timing of market reaction surrounding the observance of a holiday by investigating if there was any pattern in the returns of the stocks that could be as a result of a holiday occurring. 3.5 Time series analysis 3.5.1 Stationarity of time series A requirement for the adoption of any of the Box-Jenkins methods is that the underlying time series must be stationary. A stationary process is one whose statistical properties like the mean value and variance are constant over time. A non-stationary process, on the other hand, is one where all the conditions aforementioned are violated. A stationary time series is defined as “a time series with a constant mean, constant variance and a constant autocovariances for each given lag” (Brooks, 2008, p. 318). In a layman’s words, it can be said to be a process whose statistical properties such as the mean and the variance of the distribution do not change over time, that is, they do no depend on time. Most time series are not stationary and are said to be stochastic in nature and can lead to spurious regressions (Brooks, 2008, p. 319). An observed time series {𝑌(𝑡); 𝑡 𝜖 𝑇} is described as weakly stationary if the following properties hold: i. 𝐸[𝑌(𝑡)] = 𝜇, ∀ 𝑡 𝜖 𝑇 ii. 𝑉𝑎𝑟[𝑌(𝑡)] = 𝜎2 < ∞ iii. 𝐶𝑜𝑣 (𝑌𝑡 , 𝑌𝑡−𝑗) = 𝛾𝑗 , ∀ 𝑡 𝜖 𝑇 49 University of Ghana http://ugspace.ug.edu.gh Also, the time series is said to be strictly stationary if in addition to the properties i., ii, and iii. listed above and iv. If the joint probability distribution function of {𝑦𝑡−𝑠 , 𝑦𝑡−𝑠+1, … , 𝑦𝑡 , … , 𝑦𝑡+𝑠−1 , 𝑦𝑡+𝑠 } is independent of 𝑡 for all 𝑠.. In testing for stationarity condition of a time series, one can use partial autocorrelation function, Ljung and Box statistics and unit root tests. In the case where the time series have a unit root, those series are said not to be stationary. Over the years, there has been a variety of unit root tests discovered in most economic and financial literature. They include Dickey and Fuller (1979), Phillips and Parron (1988), Kwiatkowski, Phillips, Schant and Shin (1992), amongst other unit root tests. To investigate whether or not the variable adjusted returns for thin trading is stationary, the unit root test was carried out. This study adopted the Augmented Dickey-Fuller (ADF) unit root test which is based on the following null hypothesis and the alternative hypothesis: 𝐻0: ?̃?𝑡 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑎 𝑢𝑛𝑖𝑡 𝑟𝑜𝑜𝑡 versus 𝐻1: ?̃?𝑡 𝑖𝑠 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 or mathematically expressed: 𝐻0: ∅ = 1 against 𝐻1: ∅ < 1 using the regression: 𝑦𝑡 = ∅𝑦𝑡−1 + 𝜀𝑡 (3.4) If the 𝑝 − 𝑣𝑎𝑙𝑢𝑒 is greater than the significance level of the test, then the null hypothesis (𝐻0) is accepted and the adjusted returns for thin trading (?̃?𝑡) series is not stationary. The ADF statistics is compared with the critical value from the Fuller’s table. 50 University of Ghana http://ugspace.ug.edu.gh 3.5.2 ARMA - GARCH models ARMA models incorporating GARCH-type innovations have been widely used to analyse particularly economic and financial time series data because of their stylized properties which include leptokurtic, volatility clustering and leverage effects ((Makridakis & Hibon, 2000; Brooks, 2008, p. 380; Tolikas, 2011). The ARMA-GARCH models are basic and important because the theorems and methods obtained in these models form the basis for further inferences for more sophisticated models (Oh & Lee, 2017). The ARMA-GARCH model seems to be preferably better to adopt when testing for seasonalities than the OLS regression because ARMA-GARCH has the capacity of dealing with both autocorrelation and a time-varying variance in the dataset (Yuan & Gupta, 2014; Wasiuzzaman, 2017; Brooks, 2008, p. 386). ARMA models consisting of Autoregressive (AR) and Moving Average (MA) models is widely used to model and forecast linear time series that are considered to be autocorrelated (Greene, 2003, p. 619; Brooks, 2008, p. 224). One requirement that needs to be considered when using ARMA models is that the time series under consideration should be stationary. GARCH models have over the years received considerable amount of attention from both the academic and other stakeholders since their discovery by Engle (1982), Bollerslev (1986) and Taylor (1986). The classes of models have in the past become important and have played a vital role in financial literature and most especially in the analysis of financial time series data particularly when it has to do with analysing and forecasting volatility (Angabini & Wasiuzzaman, 2011). To achieve the objectives of this study the univariate GARCH type time series model in the conditional variance with an ARMA model specification in the conditional mean with exogenous variables was used. 51 University of Ghana http://ugspace.ug.edu.gh 3.5.3 ARMAX (p, q) – GARCH (x, y) model As stated in the previous section, this study used the ARMAX (p, q) - GARCH (x, y) to achieve its earlier stipulated objectives. An extension of the ARMA model is the ARMAX model, which is an Autoregressive Moving Average (ARMA) model with exogenous input variables, 𝑋 (Pickup, 2015, pp. 114-115). For this study the exogenous variables are the dummy variables: preholiday and postholiday. This model will enable the researcher know if the returns of the sub-periods (pre- and post-holidays) are statistically and significantly higher than the returns of the normal trading days. The ARMAX (p, q) – GARCH (x, y) are generally expressed as in the equation below: 𝑝 𝑞 𝑌𝑡 = 𝑐 + ∑ 𝛼 𝑛 𝑖=1 𝑖𝑌𝑡−𝑖 + ∑ ∑𝑗=1 𝛽𝑗𝜀𝑡−𝑗 + 𝑘=1 𝛾𝑘𝑋𝑘,𝑡 + 𝜀𝑡 , 𝜀𝑡~(0,𝜎 2) (3.5) where 𝑌𝑡 is the dependent variable, 𝑐 is the constant term, 𝛼𝑖 are the coefficients of the 𝑖 𝑡ℎ order of the AR model and 𝛽𝑗 are the coefficients of the 𝑗 𝑡ℎ order of the MA model, 𝑌𝑡−𝑖 is the lagged of the dependent variable, 𝛾𝑘 coefficient of the 𝑋𝑘,𝑡 , exogenous variable at time 𝑡, 𝑛 is the total number of 𝑋𝑘,𝑡 , 𝜀𝑡 is the error term (a white noise process) of the model, 𝑝 and 𝑞 are the order of the AR model and MA model respectively. The innovations of 𝜀𝑡 are modelled as GARCH (x, y) and illustrated below as: 𝜀𝑡 = 𝜇 + 𝜎𝑡𝑧𝑡 , (3.6) 𝑦 𝜎2 = 𝜔 + ∑𝑥 𝜔 𝜎2 + ∑ 𝜌 𝑧2𝑡 0 𝑖=1 𝑖 𝑡−𝑖 𝑗=1 𝑗 𝑡−𝑗 (3.7) 𝜔0 > 0, 𝜔𝑖 ≥ 0 and 𝜌𝑗 ≥ 0 , 𝜔𝑖 + 𝜌𝑗 < 1 (3.8) where 𝜇 is the mean value of the returns, 𝜎2𝑡 is the conditional variance based on the historical data, 𝜔𝑖 expresses how volatility responds to movements in the market (ARCH 52 University of Ghana http://ugspace.ug.edu.gh effects of the ith order of the AR model) and 𝛿𝑗 measures the persistence shocks caused by extreme values of the conditional variance (GARCH effects of the jth MA model)); According to Brooks (2008, p. 394), the GARCH (1, 1) is sufficient to capture all the volatility clustering in the data. Therefore, the study used the GARCH (1, 1) process to model the volatility present in the returns series data (?̃?𝑡) and assumed it had a 𝐺𝐿 + distribution. 3.5.4 ARMAX (p, q) model specification as applied in the study As stated in earlier sections, ARMA is often referred to as ARMAX when it includes other time series inputs. For this study, the ARMAX- GARCH model was estimated and modelled in the following form and assumed a 𝐺𝐿+ distribution (Andoh, 2009; Yuan & Gupta, 2014; Wasiuzzaman, 2017 & 2018). The equations 3.9 and 3.10 below were estimated to achieve the first and second objectives of this study respectively. Mean Equation: ?̃?𝑡 = 𝑐 + 𝛽1𝑝𝑟𝑒ℎ𝑜𝑙𝑖𝑑𝑎𝑦 + 𝛽2𝑝𝑜𝑠𝑡ℎ𝑜𝑙𝑖𝑑𝑎𝑦 + 𝜔𝑝?̃?𝑡−2 + 𝛿𝑞𝜀𝑡−𝑞 + 𝜀𝑡 (3.9) ?̃?𝑡 = 𝑐 + ∑ 11 𝑗=1 𝛽𝑗𝑝𝑟𝑒ℎ𝑜𝑙𝑖𝑑𝑎𝑦 + ∑ 22 𝑗 𝑗=12 𝛽𝑗𝑝𝑜𝑠𝑡ℎ𝑜𝑙𝑖𝑑𝑎𝑦𝑗 + 𝜔𝑝?̃?𝑡−2 + 𝛿𝑞𝜀𝑡−𝑞 + 𝜀𝑡 (3.10) where 𝑝𝑟𝑒ℎ𝑜𝑙𝑖𝑑𝑎𝑦 and 𝑝𝑜𝑠𝑡ℎ𝑜𝑙𝑖𝑑𝑎𝑦 are dummy variables that represent 1 for all pre- holiday average returns and post-holiday average returns respectively and 0 otherwise, 𝑐 is the average returns for normal trading days, 𝛽𝑗 , 𝑗 = 1,… ,22 are the average returns coefficients for either pre-holidays and post-holidays for holiday 𝑗. 3.5.5 𝑮𝑳+ distribution A random variable 𝑋 has the 𝐺𝐿+ distribution (𝑓𝑜𝑟 𝑠ℎ𝑜𝑟𝑡 𝐺𝐿+(𝜇, 𝑣2 , 𝑎, 𝑏)) if its density is given by: 53 University of Ghana http://ugspace.ug.edu.gh 𝑥−𝜇 − 𝑙𝑜𝑔𝑎 𝑎 𝑣 𝑓(𝑥) = 𝑏 𝑏+1 , −∞ < 𝑥 < ∞ , 𝑏 > 0 , 𝑥 𝜖 𝑋 (3.11) 𝑣 𝑥−𝜇− [1+𝑎 𝑣 ] where the parameters 𝜇 𝜖 (−∞,∞), 𝑣2 > 0, 𝑎 𝜖 ℝ+ and 𝑏 > 0 are the location (mean), scale, shape and skewness respectively (Andoh, 2010 & 2009; Andoh et al., 2018). According to Andoh (2009) 𝐺𝐿+ is a Type 1 Generalized Logistic distribution of log 𝑎 Balakrishman (1992) and Balakrishman and Leung (1988) and by letting 𝛼 = and 𝑎 = 𝑣 𝑒log 𝑎 the distribution is just the logistic distribution. The 𝐺𝐿+ distribution does not separately depend on 𝑣, 𝑐 and 𝑏 but on the parameters 𝛼 and 𝑏. As stated earlier, one important feature of the 𝐺𝐿+ distribution is that by adjusting at least one of the parameters, the shape and skewness of the parameters can be adjusted with ease to obtain a desired level (Andoh, 2009; Andoh 2010; Andoh et al., 2018). Andoh (2009) showed that there exists a unique 𝑣 ?̃? = exp ( √𝜑′(𝑏) + 𝜑′(1)) (3.12) 𝜎 assuming 𝑉𝑎𝑟(𝑋) = 𝜎2 and 𝐸(𝑋) = 0. 𝑣 i. 𝐸(𝑋) = 𝜇 ± [𝜑(𝑏) − 𝜑(1)] = 0 and log 𝑎 𝑣 2 ii. 𝑉𝑎𝑟(𝑋) = ( ) [𝜑′(𝑏) + 𝜑′(1)] = 𝜎2 log𝑎 Again, according to Andoh (2010), in the calculation of the GARCH volatility one common standard assumption of the Standardized 𝐺𝐿+ (𝑆𝐺𝐿+) distribution is that it has a mean of 0 and a variance of 1. Therefore, for this study and with the adoption of a 𝐺𝐿+ distribution for the GARCH model (as illustrated in the sections above) the 𝑆𝐺𝐿+ innovation was assumed. That is, it was assumed that the 𝐺𝐿+ distribution had the following properties; a 54 University of Ghana http://ugspace.ug.edu.gh mean of 0 and variance of 1. Therefore, the innovation 𝑧𝑡 in this study was assumed to follow the 𝑆𝐺𝐿+ distribution which is defined as follows: (𝜑(𝑏)−𝜑(1)) 𝑧𝑡~𝑆𝐺𝐿 + (- , 1 , ?̃?, 𝑏) (3.13) √𝜑′(𝑏)+𝜑′(1) where 𝜑 and 𝜑′ are the digamma and trigamma functions respectively and defined generally below as: 𝑑𝑛+1 𝜑(𝑛)(𝑏) = (𝑙𝑜𝑔𝛤(𝑏)) (3.14) 𝑑𝑏𝑛+1 ∞ where (𝑏) = ∫ 𝑡𝑏−1 𝑒−𝑡 𝑑𝑡 , for a positive constant b 0 Therefore, digamma function 𝜑 is the derivative of the logarithm of the gamma function (Greene, 2003, p. 928). 𝑑1 𝜑(𝑏) = (1 𝑙𝑜𝑔𝛤(𝑏)) (3.15) 𝑑𝑏 And the trigamma function is the second derivative of the logarithm function of the gamma function and it is expressed as (Greene, 2003, p. 928): 𝑑2 𝜑′ (𝑏) = ( ( ))2 𝑙𝑜𝑔𝛤 𝑏 (3.16) 𝑑𝑏 3.5.6 Maximum Likelihood Estimations (MLE) for volatility models Due to the non-linear nature of volatility the OLS is not a good method to estimate the parameters adequately and therefore this study adopted the MLE because it gives consistent estimates even if the distribution in question has a non-normal density function (Berkes et al., 2003; Andoh, 2009; Brooks, 2008). The OLS reduces the Residual Sum of Squares (RSS) because the RSS depends on the parameters in only the conditional mean and not the unconditional variance and therefore makes it inappropriate for use (Brooks, 2008). The 55 University of Ghana http://ugspace.ug.edu.gh MLE seeks to best estimate the parameters that are most likely to generate the data used for this study. The MLE are based on a numerical approximation (direct numerical integration) of the formulae. As a result, MLE is difficult and time consuming to implement on samples in modern finance. Nonetheless, there have been the development of several sophisticated softwares over the years that seek to easily compute these rigorous mathematical expressions. Therefore, using the MLE method to estimate the GARCH parameters for this study, the errors (𝑧𝑡) were assumed to follow a 𝑆𝐺𝐿 + distribution as elaborated earlier. 3.5.7 Parameter estimation Let 𝑓(𝑋, 𝜃) denote the joint density function for a vector of observations defined as: 𝑋 = (𝑋1, 𝑋2, 𝑋3, … , 𝑋𝑛) (3.17) where 𝜃 is the parameter space of 𝜇 and 𝑣. For 𝑋 = (𝑋1, 𝑋2) the conditional probability function (or likelihood function) is: 𝑓(𝑋 ,𝑋 ) 𝑓(𝑋 2 12|𝑋1) = (3.18) 𝑓(𝑋1) 𝑓(𝑋2, 𝑋1) = 𝑓( 𝑋1). 𝑓(𝑋2|𝑋1) (3.19) For 𝑋 = (𝑋1, 𝑋2 , 𝑋3) the conditional probability function (or likelihood function) is: 𝑓(𝑋 𝑋 ,𝑋 ) 𝑓(𝑋 |𝑋 , 𝑋 ) = 3, 2 13 1 2 (3.20) 𝑓(𝑋2,𝑋1) 𝑓(𝑋3 ,𝑋2, 𝑋1) = 𝑓(𝑋2, 𝑋1). 𝑓(𝑋3|𝑋1, 𝑋2) (3.21) For 𝑋 = (𝑋1, 𝑋2 , 𝑋3, 𝑋4) the conditional probability function is: 𝑓(𝑋 ,𝑋 ,𝑋 ,𝑋 ) 𝑓(𝑋4|𝑋1, 𝑋2, 𝑋3) = 4 3 2 1 (3.22) 𝑓(𝑋3,𝑋2,𝑋1) 𝑓(𝑋4, 𝑋3,𝑋2, 𝑋1) = 𝑓(𝑋3, 𝑋2, 𝑋1). 𝑓(𝑋4|𝑋1, 𝑋2, 𝑋3) (3.23) 56 University of Ghana http://ugspace.ug.edu.gh For a 𝑡 number of observations; if 𝑋 = (𝑋1, 𝑋2, 𝑋3, … , 𝑋𝑡), then the conditional probability function is given as; 𝑓(𝑋𝑡 , … , 𝑋3,𝑋2, 𝑋1) = 𝑓(𝑋3, 𝑋2, 𝑋1). 𝑓(𝑋𝑡|𝑋1, 𝑋2, 𝑋3, … , 𝑋𝑡−1) (3.24) 𝑓(𝑋| 𝜃) = ∏𝑛𝑡=(𝑝𝑉𝑞)+1 𝑓(𝑋𝑡|𝐹𝑡−1) . 𝑓(𝑋𝑝𝑉𝑞 , … , 𝑋2, 𝑋1) (3.25) where Ϝ𝑡−1 = (𝑋𝑡−1, 𝑋𝑡−2, 𝑋𝑡−3, ……… ,𝑋𝑡−(𝑝𝑉𝑞)) and 𝑛 ≫ 𝑝, 𝑛 ≫ 𝑞. The log - likelihood function is given as: 𝑙(𝜃 ⃓ 𝑋) = ∑𝑛𝑡=(𝑝𝑉𝑞)+1 log( 𝑓(𝑋𝑡⃓ Ϝ𝑡−1)) + log( 𝑓(𝑋𝑝𝑉𝑞 , … , 𝑋2, 𝑋1)) (3.26) Let 𝑙 𝑛𝑐 = ∑𝑡=(𝑝𝑉𝑞)+1 log( 𝑓(𝑋𝑡⃓ Ϝ𝑡−1)) however there is no analytical form the second term in equation 3.27. As 𝑛 → ∞ , 𝑙 − 𝑙𝑐 is neglible (See Andoh, 2010). Therefore 𝑙𝑐 is the conditional distribution of 𝑋𝑡 given its past information Ϝ𝑡−1. Now the probability density function for a 𝐺𝐿+ distributed random variable let’s say, 𝑋𝑡 with the parameter (𝜃 = 𝜇, 𝑣 ) is given by: 𝑥−𝜇 − 𝑙𝑜𝑔𝑎 𝑎 𝑣 𝑓(𝑋𝑡| 𝜃 = 𝜇, 𝑣) = 𝑏 𝑏+1 (3.27) 𝑣 𝑥−𝜇− [1+𝑎 𝑣 ] Since the 𝑋𝑡′𝑠 are independently and identically distributed (i.i.d.) variables the joint probability density function (pdf) for all the dependent variables can be expressed as a product of each density function as shown in equations (3.25). 𝑓(𝑋1, 𝑋2, 𝑋3, 𝑋4 …… , 𝑋𝑛| 𝜃 = 𝜇, 𝑣) = 𝑓(𝑋1| 𝜃 = 𝜇, 𝑣) × 𝑓( 𝑋2| 𝜃 = 𝜇, 𝑣) × 𝑓(𝑋3| 𝜃 = 𝜇, 𝑣) ×…… × 𝑓(𝑋𝑛| 𝜃 = 𝜇, 𝑣) (3.28) = ∏𝑛𝑡=1 𝑓(𝑋𝑡| 𝜃 = 𝜇, 𝑣) , 𝑡 = 1,2,…… . , 𝑛 57 University of Ghana http://ugspace.ug.edu.gh The terms on the LHS and the RHS are the joint density and the marginal densities respectively. The likelihood function is similar to the joint density which is used to estimate the parameters concerned (𝜃) is given as: 𝑥 − 𝑡 −𝜇 𝑙𝑜𝑔𝑎 𝑎 𝑣 𝐿(𝑋𝑡| 𝜃 = 𝜇, 𝑣) = ∏ 𝑛 𝑡=1 [𝑏 𝑏+1] (3.29) 𝑣 𝑥 −𝜇− 𝑡 [1+𝑎 𝑣 ] It is difficult differentiate the above equation because it is a product of n terms and since max 𝑓(𝑥) = max 𝐼𝑛 (𝑓(𝑥)) the logs of the equation above can be taken (Brooks, 2008, pp. 446). The log-likelihood function is obtained as: 𝑥𝑡−𝜇− 𝑙𝑜𝑔𝑎 𝑎 𝑣 𝑙(𝑋𝑡|𝜃 = 𝜇, 𝑣) = log [∏ 𝑛 𝑡=1 (𝑏 𝑏+1) ] (3.30) 𝑣 𝑥𝑡−𝜇− [1+𝑎 𝑣 ] 𝑥 − 𝑡 −𝜇 𝑙𝑜𝑔𝑎 𝑎 𝑣 𝑙(𝑋𝑡|𝜃 = 𝜇, 𝑣) = ∑ 𝑛 𝑡=1 [log (𝑏 ) + log( 𝑏+1)] (3.31) 𝑣 𝑥− 𝑡−𝜇 [1+𝑎 𝑣 ] 𝑙(𝑋𝑡|𝜃 = 𝜇, 𝑣) log(𝑎) 𝑥𝑡−𝜇 𝑥𝑡−𝜇 𝑏+1 − − = ∑𝑛𝑡=1 {log(𝑏) + log ( ) + log (𝑎 𝑣 ) − log [([1 + 𝑎 𝑣 ]) ]} (3.32) 𝑣 𝑙(𝑋𝑡| 𝜃 = 𝜇, 𝑣) = 𝑥𝑡−𝜇 ∑𝑛 𝑥 −𝜇 𝑡=1 {log(𝑏) + log(log(𝑎)) − log (𝑣) − 𝑡 log (𝑎) − (𝑏 + 1) log (1 + 𝑎− 𝑣 )} 𝑣 (3.33) 𝑙(𝑋𝑡| 𝜃 = 𝜇, 𝑣) x -μ n xt-μ t = ∑t=1 { ( - log b) + log(a) [log(1)- ] - log(v) - (b+1) log (1+a v )} (3.34) v 58 University of Ghana http://ugspace.ug.edu.gh 𝑙(𝑋𝑡| 𝜃 = 𝜇, 𝑣) = 𝑥 −𝜇 𝑥𝑡−𝜇 ∑𝑛𝑡=1 {log(𝑏) + (log1 − 𝑡 ) log(𝑎) − log(𝑣) − (𝑏 + 1) × log (1 + 𝑎− 𝑣 ) } (3.35) 𝑣 𝑙(𝑋𝑡| 𝜃 = 𝜇, 𝑣) = 𝑥 −𝜇 𝑥− 𝑡 −𝜇 ∑𝑛𝑡=1 {log(𝑏) − 𝑡 log(𝑎) − log(𝑣) − (𝑏 + 1) log (1 + 𝑎 𝑣 ) } (3.36) 𝑣 Therefore the negative conditional distribution of 𝑋𝑡 given its past information Ϝ𝑡−1 (𝑙𝑐) and neglecting the constant in the log-likelihood function (see Andoh, 2010) we have: 𝑥𝑡+𝜇 𝑙 = − ∑𝑛 𝑥 +𝜇 𝑐 𝑡=(𝑝𝑉𝑞)+1 [−log𝑏 + log𝑣 + ( 𝑡 ) log𝑎 + (𝑏 + 1) log (1 + 𝑎−( )𝑣 )] (3.37) 𝑣 Given the GARCH model as stated in equation (3.6) and assuming that + (𝜑(𝑏)−𝜑(1))𝜀𝑡~ 𝑆𝐺𝐿 (− , 1 , ?̃?, 𝑏) then its probability density function is given as: √𝜑′(𝑏)+𝜑′(1) 𝜎 (𝜑(𝑏)−𝜑(1)) 𝑓(𝑧 + 𝑡𝑡|𝐹𝑡−1) = 𝑆𝐺𝐿 (− , 𝜎𝑡 , ?̃?, 𝑏) (3.38) √𝜑′(𝑏)+𝜑′(1) and 𝑛 (𝜑(𝑏)−𝜑(1)) 𝑙𝑐 = −∑𝑡=(𝑝𝑉𝑞)+1 − log𝑏 + (𝑥𝑡 − ) log ?̃? + (𝑏 + 1) log 1 +√𝜑′(𝑏)+𝜑′(1) [ ( (𝜑(𝑏)−𝜑(1)) −(𝑥𝑡− ) √𝜑′(𝑏)+𝜑′(1) ?̃? (3.39) )] The −𝑙𝑐 is differentiated and equated to 0 to solve for the global maximum values of the parameters concerned. 59 University of Ghana http://ugspace.ug.edu.gh Hence the log-likelihood function adopted for this study is written as: 𝑥 −𝑙𝑐 = ∑ 𝑛 [− log𝑏 + log𝜎 (𝜃) + 𝑡 √𝜑′ ′𝑡=(𝑝𝑉𝑞)+1 𝑡 (𝑏) + 𝜑 (1) + (𝜑(𝑏) − 𝜑(1)) +𝜎𝑡(𝜃) 𝑥𝑡 𝜑(𝑏)−𝜑(1)− 𝜎𝑡(𝜃) ′ ′ (𝑏 + 1) log(1 + exp(√𝜑′(𝑏) + 𝜑′( ) √𝜑 (𝑏)+𝜑 ( ) 1 ) 1 ] (3.40) for any admissible value of 𝜃 of the parameter spaceΘ. The values of the parameters are determined by differentiating the log-likelihood function and finding their maximum values that are likely to produce the data used. 3.5.8 GARCH model specification as applied in the study The volatility in this study was modelled using a GARCH (1, 1) model and it is expressed below as: Variance Equation (Andoh, 2010): 𝜀𝑡 = 𝜎𝑡𝑧𝑡 (3.41) + (𝜑(𝑏)−𝜑(1)) 𝑧𝑡|𝐹𝑡−1 ~𝑆𝐺𝐿 (- , 1, ?̃?, 𝑏) (3.42) √𝜑′(𝑏)+𝜑′(1) 𝜎𝑡(𝜑(𝑏)−𝜑(1)) 𝜀 +𝑡|𝐹𝑡−1 ~𝑆𝐺𝐿 (- , 𝜎 2 𝑡 , ?̃?, 𝑏) (3.43) √𝜑′(𝑏)+𝜑′(1) ?̃? = exp ( √𝜑′(𝑏) + 𝜑′(1)) (3.44) 𝜎2 = 𝛼 + 𝛼 2 2𝑡 0 1𝑧𝑡−1 + 𝜌1𝜎𝑡−1 (3.45) 𝛼0>0, 𝛼1 ≥ 0, 𝜌1 ≥0, 𝛼1 + 𝜌1 < 1 (3.46) 60 University of Ghana http://ugspace.ug.edu.gh Maximum Likelihood function: 𝑥 −𝑙𝑐 = ∑ 𝑛 𝑡 ′ ′ 𝑡=(𝑝𝑉𝑞)+1 [− log𝑏 + log𝜎𝑡(𝜃) + √𝜑 (𝑏) + 𝜑 (1) + (𝜑(𝑏) − 𝜑(1)) +𝜎𝑡(𝜃) 𝑥𝑡 𝜑(𝑏)−𝜑(1)− 𝜎𝑡(𝜃) ′ ′ (𝑏 + 1) log(1 + exp(√𝜑′(𝑏) + 𝜑′(1) √𝜑 (𝑏)) +𝜑 (1) ] (3.47) 3.5.9 ARMAX-GARCH model testing There are a number of approaches that are adopted to determine the right order for an ARMA model. Some of these commonly used approaches are the autocorrelation coefficients (acf), partial autocorrelation coefficient (pacf) and the information criteria (Brooks, 2008, pp. 230- 232). The information criterion is based on a concept that an additional independent variable leads not only to an increase in the goodness of good fit but also an increase in the accuracy of the forecast. Therefore, the information criteria impose a penalty on the lost degree of freedom because an addition of one or more independent variables(s) to the model increases uncertainty as a result of changes in the degree of freedom. There are approximately 3 types of information criteria; Akaike Information Criteria (AIC), the Schwartz Bayesian Information Criteria (SBIC) and Hannan-Quinn Information Criteria (HQIC) (Brooks, 2008, p. 233). Akaike Information Criteria (AIC) is defined as: 2(𝑝+𝑞+1) AIC = ln(𝜎2̂) + (3.48) 𝑇 where 𝜎2̂ is the variance of the residuals and 𝑇 is the sample size. 61 University of Ghana http://ugspace.ug.edu.gh Schwartz Bayesian Information Criteria (SBIC), imposes a smaller penalty on the added regressors as compared to the (AIC)’s and it is defined as: 2(𝑝+𝑞+1) 𝑆𝐵𝐼𝐶 = ln(𝜎2̂) + ln(𝑇) (3.49) 𝑇 where 𝜎2̂ is the variance of the residuals and 𝑇 is the sample size. Hannan-Quinn Information Criterion (HQIC), imposes a smaller penalty on the added regressors as compared to both AIC and BIC. It is defined as: 2(𝑝+𝑞+1) 𝐻𝑄𝐼𝐶 = ln(𝜎2̂) + ln(𝐼𝑛(𝑇)) (3.50) 𝑇 where 𝜎2̂ is the variance of the residuals and 𝑇 is the sample size. 3.6 Risk metrics In finance, the return of a financial asset could be as a result of the its associated risk (volatility) (Brooks, 2008, p. 410). Hence in dealing with such situations the Value at Risk (VaR) and the Expected Shortfall (ES) and also referred to as the Conditional Value at Risk (CVAR) were used to measure the risk associated with the holiday effect on the Ghana Stock Exchange (GSE). Some examples of risks that have been discovered in financial literature include operational risk, credit risk, liquidity risk, political risk and market risk. The VaR and the ES are both risk measurements accepted in and by the financial community. The VaR is a one of the most widely used risk measures, recommended for the calculations of market risk and mentioned in Basel II (Obi & Sil, 2013). For example, Andoh, (2010) applied GARCH in VaR calculation for some selected German stocks to calculate the risk of these stocks. The VaR is defined as the worst possible loss over a given period of time and at a given significance level. The VaR of a portfolio at 𝛼 significance level such that 62 University of Ghana http://ugspace.ug.edu.gh 𝛼 {0 ≤ 𝛼 ≤ 1} is the least number 𝜁such that the probability of the worst possible loss on the portfolio being more than 𝜁 is at most (1 − 𝛼). This is mathematically expressed as: 𝑉𝑎𝑅𝛼 = inf{ 𝜁 ∈ ℝ ∶ 𝑃(𝜗 > 𝜁) ≤ 1 − 𝛼} (3.51) In a lay man’s statement, the Value at Risk at significance level could be said to the 𝛼 - quantile of the loss distribution. The Expected Shortfall (ES) or CVaR has recently been recommended by Basel III to replace the VaR because of the problems that have arisen from its use. The VaR is incapable of capturing behaviours that mostly are observed at the tail of the distribution of the risk. The CVaR was proposed by Artzner, Delbaen, Eber and Heath (1999) and it estimates the potential size of the loss that exceeds the VaR. According to Artzner et al. (1999) the VaR is not a coherent risk measure because it does not satisfy the subadditivity axiom (see proof in Danielson, 2011, p. 82). The CVaR is expressed mathematically as (Göb, 2011): 𝐶𝑉𝑎𝑅 = (1 − 𝛼)−1 1 𝛼 ∫ 𝑉𝑎𝑅𝑢 𝑑𝑢 (3.52) 𝛼 Again, the value of the CVaR risk measure is always higher than the value of the VaR risk measure, simply, because the former is larger than the later by the average excess of all losses exceeding VaR (Danielson, 2011, p. 87). 3.7 Software used for the estimation of the models The study used E-Views 9 to estimate the OLS regression and same was used to conduct the residual diagnostic tests. MATLAB 2017a was used to estimate the ARMAX-GARCH model because of the nature of the model (especially the distributional properties). Also, the study used Kevin Sheppard MFE toolbox for the ARMAX estimations and Microsoft Excel 2016 to code the dummy variables. 63 University of Ghana http://ugspace.ug.edu.gh CHAPTER FOUR EMPIRICAL RESULTS AND ANALYSIS OF FINDINGS 4.1 Introduction This chapter entails an analysis and interpretation of the data collected. As noted earlier the data set was collected from the Ghana Stock Exchange. This chapter shows a descriptive analysis of the variables and presents findings relating to the effect of holidays on the stock market in Ghana and those relating to the other objectives stated in the study earlier. 4.2 Time series Source: GSE, 2007-2016 Figure 4.1: Time plot of daily closing prices of GSE index- 2007-2016 A plot of the time series consisting of the daily closing prices (𝑃𝑡, the raw data) is shown in Fig. 4.1 which indicates an upward trend from the years 2007 to late 2008. In 2011, there was 64 University of Ghana http://ugspace.ug.edu.gh a massive reduction of the price index from approximately 7000’s to about 1000 due to the change of the index from the GSE-All Shares to the GSE-CI index (a capitalization index with a base value of 1000). The plot shows that the variance of 𝑃𝑡 seems not to be stable over time and it is more volatile especially between the years 2010 and 2011. These observations imply that a log transformation of the closing prices is appropriate to ensure that the data is stationary. Time series plot of the re-scaled log returns (𝑅𝑡) and the adjusted for thin trading returns (?̃?𝑡) of the GSE-All Shares and the GSE-CI index are shown in Figure 4.2 and Figure 4.3 respectively. The volatility of both returns are not too different from each other. The series are stationary and volatile; there is more turbulence in the later year of 2013 and relatively quiet periods at the beginning of 2007. The periods with high or low volatility tend to appear in a cluster as a result of volatility clustering. Source: Researchers calculations, 2018. Source: Researcher, 2018. Figure 4.2: Time series plot of daily log returns (Jan. 2007- Dec. 2016) 65 University of Ghana http://ugspace.ug.edu.gh Source: Researcher’s calculations, 2018. Figure 4.3: Time series plot of daily adjusted for thin trading returns (Jan. 2007- Dec. 2016) 4.3 Descriptive statistics for the adjusted thin trading stock returns A summary of the descriptive statistics for adjusted thin trading stock returns for the entire sample period January, 2007 to December, 2016 used in the study is presented in the table below. The table shows the mean, maximum and minimum returns as well as the number of observations for pre-holiday, post-holiday and normal days, skewness, standard deviations and the Jarque-Bera statistics of the thin trading adjusted stock returns. 66 University of Ghana http://ugspace.ug.edu.gh Table 4.1: Descriptive statistics of thin trading adjusted stock returns ADJUSTED RETURNS Mean -3.63E-17 Maximum 10.50740 Minimum -11.36276 Std. Dev. 0.837620 Skewness -0.723905 Kurtosis 40.38739 Jarque-Bera 144307.7 Probability 0.000000 Sum -7.77E-14 Sum Sq. Dev. 1735.074 Observations 2474 Source: GSE, 2007-2016 According to the descriptive statistics of the daily return series adjusted for thin trading as shown in the Table 4.1 above, the total number of observations is 2474. The average return of the series is -3.63E-15%, implying negative skewness which shows that the distribution of the return of the series is asymmetric. The maximum and minimum average returns are 10.50 basis point and -11.36 basis point respectively. The return series has a high kurtosis greater than 3 which implies that the return is fat tailed. Return is leptokurtic with excess positive kurtosis of approximately 37.39 above the normal distribution, contributing to the presence of big shocks in the series with either type of sign. The distribution has a corresponding high Jarque-Bera statistics coefficient and the null hypothesis of normality is rejected at 1% level. The daily standard deviation of 83.76% is also high; this indicates a high variability in the Ghana Stock Exchange Composite returns adjusted for thin trading series. The high daily variation in the return series is an indication that it is probably and highly risky to invest in such an emerging market. These characteristics are also consistent with findings in emerging markets by Mlambo et al. (2007). 67 University of Ghana http://ugspace.ug.edu.gh 4.4 Residual diagnostic tests 4.4.1 Autocorrelation test Gujarati and Porter (2009, p. 472) stated that one critical problem associated with the application of the Durbin-Watson test to test the presence of serial correlation amongst the error terms was that it assumes that the regressors are non-stochastic (values are fixed in repeated sampling). They acknowledged that this assumption was usually difficult to maintain in most econometric models involving time series data and made referenced to Hayashi (2000) who was of the view that the Durbin-Watson test was not appropriate for time series data and suggested other tests such as the Breusch-Godfrey test. Most financial time series naturally do not exhibit the characteristic of serially independence (the situation where the residuals are not correlated), however, this is more consistent in the case of a series consisting of stock market returns computed from an emerging market. The explanation to this phenomenon could be the presence of thin trading (non-trading and non-synchronous trading) which is obviously one of the major characteristics of an emerging market. The Breusch-Godfrey Serial Correlation LM test is used to test the presence of serial dependence structure in the daily return series. From the Breusch-Godfrey Serial Correlation LM test, the null hypothesis that there is no serial correlation in the residuals is rejected. This result presents a significant serial dependence composition in the return series. The results from the serial autocorrelation test is shown in the Table 4.2 below: Table 4.2: Results of Breusch-Godfrey serial correlation LM test Breusch-Godfrey Serial Correlation LM Test: F-statistic 5.912597 Prob. F(36,2433) 0.0000 Obs*R-squared 198.9478 Prob. Chi-Square(36) 0.0000 Source: Researcher’s calculation, 2018. 68 University of Ghana http://ugspace.ug.edu.gh Hence from the result of the serial autocorrelation test this distribution exhibits serial correlation with p-value of 0 which is lower than 1% level of significance. 4.4.2 Heteroscedasticity test The test for heteroscedasticity conducted on the residuals in the return series shows a significant presence of heteroscedasticity amongst the residuals. The 1% significance level is an indication that there is a dependence structure in the first and higher moments and therefore the ARCH-GARCH models can be adopted. This result is corroborated by the presence of the high volatility (high standard deviation of approximately 83.76%) and kurtosis (approximately 40.39) in the return series. The rejection of the null hypothesis that the variance is constant implies that the family of ARCH-GARCH models should be used to estimate volatility and forecast. The summary of the test for heteroscedasticity is shown in Table 4.3 below: Table 4.3: Results of the test for heteroscedasticity (White test) Heteroskedasticity Test: White F-statistic 102.3326 Prob. F(3,2469) 0.0000 Obs*R-squared 273.4893 Prob. Chi-Square(3) 0.0000 Scaled explained SS 5795.756 Prob. Chi-Square(3) 0.0000 Source: Researcher’s calculation, 2018 4.4.3 Stationary test or unit root test In situations that involve modelling and analysing financial time series, stationary time series data is preferred to data that is non-stationary. This condition is preferred because if a time series data is stationary then the shocks to the system will gradually die out as time changes. The test for stationarity is performed on the daily return for thin trading series using the Augmented Dickey-Fuller unit root. The Augmented Dickey-Fuller (ADF) unit root test was performed considering 3 scenarios; allowing only for an intercept, allowing for both an intercept and deterministic trend and finally allowing for neither the intercept nor trend. In all 69 University of Ghana http://ugspace.ug.edu.gh three cases and for the whole sample period from 2007 to 2016, the results of Augmented Dickey-Fuller for a unit root for the GSE exhibit the rejection of unit root in the series at a significant level of 1% and conclude that the return series are stationary and do not contain a unit toot. This is as a result of the continuous compounded calculations used to compute the returns. The results of the ADF are shown in the Table 4.4 below and the rest in the Appendix B. Table 4.4: Results of Augmented Dickey-Fuller test with intercept Stationarity test: ADF with intercept t-Stat istic Pro b.* ADF test statistic -12.38884 0.0000 Test critical values: 5% level -2.862511 *MacKinnon (1996) one-sided p-values Source: Researchers’ calculations, 2018. In conclusion, the dataset used for this study showed existence of heteroscedasticity, serial correlation and non-normality amongst the error terms. However, the dataset is stationary and this is an indication that the ARMA model should be adopted for the analysis in this research as proposed by Yuan and Gupta (2014). 4.4.4 Model specification for ARMAX model To determine the order that will best fit the data in this study the information criteria was used. The results are shown in Table 4.5 A) and Table 4.5 B) below: 70 University of Ghana http://ugspace.ug.edu.gh Table 4.5: Model Specification for ARMAX (p, q) model. Table 4.5 A) P 0 0 0 0 0 1 1 1 1 1 Q 0 1 2 3 4 0 1 2 3 4 Adj T 2476 2476 2476 2476 2476 2475 2475 2475 2475 2475 T 2476 2476 2476 2476 2476 2476 2476 2476 2476 2476 K 2 2 2 2 2 2 2 2 2 2 AIC -0.402 -0.402 -0.422 -0.405 -0.418 -0.402 -0.401 -0.422 -0.405 -0.418 HQC -0.400 -0.398 -0.419 -0.414 -0.414 -0.398 -0.397 -0.418 -0.401 -0.413 SBIC -0.395 -0.392 -0.413 -0.395 -0.408 0.392 -0.389 -0.410 -0.393 -0.406 Table 4.5 B) P 2 2 2 2 2 3 3 3 3 3 Q 0 1 2 3 4 0 1 2 3 4 Adj T 2474 2474 2474 2474 2474 2473 2473 2473 2473 2473 T 2476 2476 2476 2476 2476 2476 2476 2476 2476 2476 K 2 2 2 2 2 2 2 2 2 2 AIC -0.429 -0.429 -0.441 -0.431 -0.436 -0.406 -0.406 -0.425 -0.408 -0.420 HQC -0.426 -0.425 -0.437 -0.427 -0.432 -0.402 -0.402 -0.421 -0.403 -0.416 SBIC -0.420 -0.417 -0.429 -0.420 -0.425 0.396 -0.394 -0.414 -0.396 -0.408 Source: Researcher’s calculations, 2018 using MatlabR2017a. The Table 4.5 A) and Table 4.5 B) show the information criteria associated with the various orders (p, q) for the ARMAX model. For example, the order (1, 0) had an AIC value of -0.402, order (1, 4) had a SBIC value of -0.406 and order (3, 3) had an HQC value of -0.403. The order that best fits the data for the ARMAX model is order (2, 2) (the shaded area) because it recorded the lowest information criteria. 4.5 Event study An event date is defined in this study as the day on which the holiday occurred during the period of study. An event window around each holiday was centred such that there was the 1 day before and 1 day after a holiday event window, and 2nd day before and 2nd day after a holiday event window till the 8th day before and 8th day after a holiday event window. The creation of these different event windows was to enable the study determine if there were any abnormal price reactions following shortly after or leading up to the occurrence and observance 71 University of Ghana http://ugspace.ug.edu.gh of a holiday. The estimation window was created for each defined event window. The OLS regression was run and the results are reported in Appendix A where they show that there is a positive pre-holiday and post-holiday effect for all the days considered in the study. These results were, however, ignored because they failed the normality test as shown in descriptive statistics table; Table 4.1. The study, thus, focused more on the ARMAX-GARCH results because of the non-normality of the error terms in the data and the presence of heteroscedasticity and serial correlation. Based on the regression model stated in equation 3.26, the ARMAX model was run and the results presented in Table 4.6. Table 4.6: ARMAX regression results (pre- and post-holiday effects) DAY WINDOW VARIABLE ±1 ±2 ±3 ±4 ±5 ±6 ±7 ±8 C 0.0002 -0.0003 -0.0007 0.0002 0.0011 0.0002 0.0002 0.0000 (0) (0) (0) (0) (0) (0) (0) (0) AR(2) 0.8411** 0.8426** 0.5406** 0.8022** 0.850** 0.8366** 0.8142** 0.8401** (0.00106) (0.0009) (0.0073) (0.0011) (0.0011) (0.0036) (0.0009) (0.0009) PRE -0.261** 0.0229** -0.565** -0.270** -0.098** 0.0380 0.1939** 0.1359** (0.0684) (0.0051) (0.0408) (0.0273) (0.0178) (0.0393) (0.0096) (0.0228) POST 0.0562** 0.0363** -0.098** -0.030** 0.1892** 0.0762 0.0894** 0.0104 (0.0233) (0.0051) (0.0120) (0.0093) (0.0089) (0.0089) (0.0098) (0.0173) MA(2) -0.711** -0.714** -0.376** -0.664** -0.728** -0.707** -0.695** -0.714** (0.0014) (0.0012) (0.0075) (0.0013) (0.0015) (0.0015) (0.0013) (0.0013) SEregression 0.8142 0.8143 0.8026 0.8093 0.8130 0.8141 0.8128 0.8138 Log - - - - - - - - Likelihood 3.00E+03 3.00E+03 2.97E+03 2.99E+03 3.00E+03 3.00E+03 3.00E+03 3.00E+03 AIC -0.4090 -0.4089 -0.4379 -0.4211 -0.4120 -0.4094 -0.4126 -0.4101 HQC -0.4047 -0.4047 -0.4336 -0.4168 -0.4078 -0.4051 -0.4083 0.4058 SBIC -0.3972 -0.3972 -0.4261 -0.4094 -0.4003 -0.3977 -0.4009 -0.3983 Notes: Standard Errors are in parentheses. **- 5% Significance level 𝜎 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑆. 𝐸. = 𝑡 − 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = √𝑛 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑟𝑟𝑜𝑟 Source: Researcher’s calculations, 2018 using MatlabR2017a. 72 University of Ghana http://ugspace.ug.edu.gh A cursory look at Table 4.6 above, shows abnormal returns for different days within the event window of interest. The regression results show that existing investors on the GSE have over the 10-year period of the study exhibited diverse trading patterns with regards to the observance and celebration of a holiday. For instance, on the 8th day before a holiday, investors are seen to have engaged in substantially higher trading activities (average returns of about 13.59%). Again, approximately 19.39% was recorded on the 7th day before a holiday and represents the day on which the most trading activities occurred. However, from the 1st day to the 5th day before a holiday there was a major reduction of trading activities with some fluctuations between these days. It is observed that out of the 16 days considered, 6 had negative returns with pre-holiday days having 4 which is the highest and represents about 50% of the observations in that sub-period (pre-holiday). However, a closer analysis of this same table reveals that the GSE documented both pre-holiday and post-holiday effects. Since, for most of the days, the returns from pre-holidays or post- holidays are much higher than those of other normal trading days. These results could be an indication that on the whole, the GSE is informationally inefficient as suggested by Dodd and Gakhovich (2011). Pre-holiday effects occurred on the following windows; the 2nd day, the 7th day and the 8th day before a holiday because the other 5 windows either recorded lower returns (in comparison to returns of normal trading days) or had results that were not significant. Also, post-holiday effects occurred on all windows except on the 3rd day, 4th day, 6th day and 8th day after a holiday. It is worth noting that despite the significant results recorded in windows such as (+1 day), (-3 days, +3 days), (-4 days, +4 days) and (+5 days), there were neither a pre- holiday effects nor post-holiday effects. This is because during these windows the average returns were lower than that of the normal trading days. On the occasion where the average returns were positive, it implied that the market experienced greater returns. Conversely, the market underperforms when it recorded negative returns. 73 University of Ghana http://ugspace.ug.edu.gh Hence, the highest average return of 19.39% for the period; pre-holiday days which occurred on the 7th day before a holiday may be due to the euphoria associated with holidays, meaning investors on the Ghana Stock Exchange currently tend to trade more on the 7th day before a holiday, in anticipation of the holiday as suggested by Gama and Vieira (2013). Again, the principle of demand and supply where excess demand lead to increase in price level may explain the phenomenon observed on the 5th day after a holiday and the post-holiday effects recorded the highest average returns of about 18.92%. Another explanation for these observations may be as a result of buy-sell strategies as iterated by Meneu and Pardo (2004), where investors are just willing to buy before a holiday and buy after a holiday. These findings are not consistent with the findings of Ariel (1990) and Dodd and Gakhovich (2011) who found holiday effects occurring on one day before and one day after a holiday in the US and some selected Central and Eastern European countries respectively. These findings as discussed above were significant at 5% significance level. Comparing the returns for the 5th day and the 7th day (these days recorded the highest return as shown in Table 4.6), the 7th day however, had the least information criteria (AIC, HQIC and SBIC) which is an indication that it is the best model fit for this data set in this study. Hence, to achieve the rest of the objectives stated in this study, the holiday effect on the GSE occurs on the 7th day before and 7th day after a holiday. The pre-holiday effect discovered supports the results of Alagidede (2013), Ariel (1990) and Dodd and Gakhovich (2011). Again, the positive post-holiday effect observed in the results, was the same observation discovered by Dodd and Gakhovich (2011) which is a contradiction to studies by Ariel (1990) who opined that usually investors after a holiday open their short selling positions which leads to lower and sometimes negative post-holiday returns. This implies that currently, investors on the GSE trade more on the 7th day before and the 7th day after a holiday. During this period the average returns before a holiday are about 19.39% and 8.94% after a holiday. 74 University of Ghana http://ugspace.ug.edu.gh Additionally, the significant pre- and post-holiday effects suggests that investors can take advantage of this anomaly and trade before or after a holiday on the GSE. For instance, on the (+7, -7) window the pre-holiday return is positively significant and about 969.501 times higher than the average return for normal trading days. The post-holiday returns are 4472 times higher than the average returns for normal trading days. Finally, the null hypothesis that there is no significant difference between the average returns of pre-holidays (post-holidays) and the average returns of normal trading days is rejected at 5% significance level. 1 𝑝𝑟𝑒ℎ𝑜𝑙𝑖𝑑𝑎𝑦 𝑟𝑒𝑡𝑢𝑟𝑛 Based on results from Table 4.6, 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑟𝑎𝑑𝑖𝑛𝑔 𝑑𝑎𝑦 𝑟𝑒𝑡𝑢𝑟𝑛 2 𝑝𝑜𝑠𝑡ℎ𝑜𝑙𝑖𝑑𝑎𝑦 𝑟𝑒𝑡𝑢𝑟𝑛Based on results from Table 4.6, 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑟𝑎𝑑𝑖𝑛𝑔 𝑑𝑎𝑦 𝑟𝑒𝑡𝑢𝑟𝑛 75 University of Ghana http://ugspace.ug.edu.gh 4.6 Specific holiday Table 4.7: ARMAX regression results for specific holidays 𝒕 - Variables Coefficient Std. Error Statistics C 0.0003 0 0 AR(2) 0.8215** 0.0039 211.2864 NYPRE 0.4894 1.1576 0.4228 NYPOST -0.2677 1.4057 -0.1904 INDEPRE 0.2755 0.5114 0.5387 INDEPOST 0.1654 0.4033 0.4101 EASTERPRE 0.1589 0.8166 0.1946 EASTERPOST 0.1226 0.2949 0.4157 WORKERPRE 0.4393 0.2358 1.8627 WORKERPOST 0.4793** 0.2413 1.9864 AUPRE 0.5168 0.2978 1.7353 AUPOST -0.0453 0.0957 -0.4736 REPUPRE 0.1753 0.1768 0.9916 REPUPOST -0.025 0.0551 -0.4534 FITRPRE 0.1771 0.2055 0.8618 FITRPOST -0.1375 0.8056 -0.1707 ADHAPRE 0.5874 0.4449 1.3202 ADHAPOST -0.0221 0.0514 -0.4301 FARMERPRE 0.9374** 0.2223 4.2167 FARMERPOST 0.4615** 0.1044 4.4224 XMASPRE 0.0794 0.3298 0.2408 XMASPOST 0.0075 0.0283 0.2652 FOUNDERPRE -0.1498 0.3531 -0.4242 FOUNDERPOST -0.2566 0.2384 -1.0764 MA(2) -0.6968** 0.0046 -151.9870 Notes: **- 5% Significance level Source: Researcher’s calculations, 2018 using MatlabR2017a. Abbreviations: PRE- PRE-HOLIDAY EFFECTS, POST- POST-HOLIDAY EFFECTS. NYPRE- NEW YEAR PRE, NYPOST-NEW YEAR POST, INDEPRE- INDEPENDENCE PRE, INDEPOST- INDEPENDENCE POST, AUPRE- AFRICAN UNION DAY PRE, AUPOST AFRICAN UNION DAY POST, REPUPRE- REPUBLIC DAY PRE, REPUPOST-REPUBLIC DAY POST, FITRPRE- EID-IL-FITR PRE, EID-IL-FITR POST, ADHAPRE- EID-AL-ADHA PRE, ADHAPOST- EID-AL-ADHA POST, FARMERPRE- FARMERS DAY PRE, FARMERPOST- FARMERS DAY POST, XMASPRE- CHRISTMAS PRE, XMASPOST- CHRISTMAS POST, FOUNDERPRE- FOUNDERS DAY PRE, FOUNDERPOST- FOUNDERS DAY POST 76 University of Ghana http://ugspace.ug.edu.gh The results in Table 4.7 suggest that Farmers day holiday celebrated in Ghana is responsible for the pre-holiday effect observed in Table 4.6. Again, both the Farmers day holiday and the Workers day holiday are contributing to the post-holiday effects documented in Table 4.6. These results show that there is some evidence of the existence of abnormally high returns on the 7th day before Farmers day of about 93.74% and on the 7th day after both Farmers day and Workers day of about 46.15% and 47.93% respectively. Again, these returns are significantly different from zero at 95% confidence level. Generally, there seems to be insufficient evidence for individual holidays generating significant returns in this study. Out of the 11 holidays considered in this study only two exhibit significant results. Even holidays that are regarded as common such as the New Year and Christmas holidays did not exhibit any significant results, contrary to Dodd and Gakhovich (2011). The alternate hypothesis that there is a significant difference amongst the average pre and post-holidays returns for each holiday is not rejected at a significance level of 5%. The holiday effect on the GSE is further examined to substantiate if investors are influenced by strictly Ghanaian-specific observed holidays or non-Ghanaian specific holidays. Ghanaian holidays for the purpose of this study are holidays that are celebrated uniquely and only recognized in Ghana. They include the Independence day, Republic day, Farmers day and Founders day holidays. From the results on the Table 4.8 A), both Ghana- specific holidays and Non-Ghanaian specific holidays contribute to the holiday effects observed in Table 4.6. Again, the results from Table 4.8 B) show that there are significant and positive pre-holiday and post-holiday effects for Ghanaian holidays and only a positive and significant pre-holiday return for Non-Ghanaian holidays. However, the average return for days before Ghanaian holidays are greater than the other 3 categories, an indication that there are more trading activities in the market during such periods. 77 University of Ghana http://ugspace.ug.edu.gh Table 4.8: ARMAX regression results for Ghana specific holidays and non-Ghana specific holidays A) GENERAL STD. 𝒕- VARIABLES COEFFICIENT ERROR STATISTICS C 0.0003 0.0000 0.0000 AR(2) 0.8216** 0.0011 772.2915 GH 0.1382** 0.0110 12.6017 NONGH 0.0571** 0.0127 4.4957 MA(2) -0.6893** 0.0014 -480.0904 B) PRE AND POST STD. 𝒕- VARIABLES COEFFICIENT ERROR STATISTICS C 0.0004 0.0000 0.0000 AR(2) 0.8192** 0.0011 743.9250 GHPRE 0.1817** 0.0232 7.8348 GHPOST 0.0793** 0.0204 3.8870 NONGHPRE 0.086** 0.0110 7.8412 NONGHPOST 0.0213 0.0275 0.7744 MA(2) -0.6857** 0.0014 -472.9686 Notes: **- 5% Significance level NONGH- NON- GHANA SPECIFIC HOLIDAY GH- GHANA SPECIFIC HOLIDAY Source: Researcher’s calculations, 2018 using MatlabR2017a. 4.7 Risk measures As opined by Andoh (2010) one interesting property of the 𝐺𝐿+ distribution is the malleable nature of the skewness and shape parameters. This is important because of the empirical features of assets returns (such as leptokurtic- fatter tails). With the use of the 𝐺𝐿+ distribution, one is able to choose the appropriate parameter that will appropriately represent the true nature of the distribution of the data and this is done by adjusting at least one of the parameters and in this case the parameter of the skewness represented by 𝑏. The options for a suitable 𝑏 parameter for this study were run and shown in the Table 4.9. 78 University of Ghana http://ugspace.ug.edu.gh Table 4.9: VaR estimates for thin trading adjusted returns with possible asymmetry in the innovations PARAMETERS 𝜔0 𝜔1 𝜌1 0.1 0.23 0.60 LEVEL OF 𝛼% VaR UNDERLYING 0.05 0.025 0.01 DISTRIBUTION 𝑆𝐺𝐿+(0.5) 0.0784 0.0574 0.0396 𝑆𝐺𝐿+(0.9) 0.0651 0.0440 0.0222 𝑆𝐺𝐿+(0.9999) 0.0630 0.0416 0.0214 𝑆𝐺𝐿+(5.1) 0.0505 0.0255 0.0125 𝑆𝐺𝐿+(10.1) 0.0501 0.0246 0.0121 𝑆𝐺𝐿+(11.1) 0.0497 0.0246 0.0121 𝑆𝐺𝐿+(12.1) 0.0497 0.0246 0.0117 Note :() are the parameter choice of the skewness (𝑏) Source: Researcher’s calculations, 2018 using MatlabR2017a. In Table 4.9, the estimate for the level where the skewness, 𝑏 = 12.1 is preferred because it has the closest values for the various levels of VaR at 5%, 2.5% and 1%. Also, the non- negativity and stationarity assumptions were adhered to. Figure 4.4 shows 5% VaR estimates (dashed lines) as well as the VaR exceedances (dotted lines) of the GARCH (1, 1) process 𝜎2𝑡 = 0.1 + 0.23𝑧 2 𝑡−1 + 0.60𝜎 2 𝑡−1 with a 𝐺𝐿 + innovation 79 University of Ghana http://ugspace.ug.edu.gh for the period 2007-2016. The exceedances (dotted line) are the areas where the VaR exceeded the 5% VaR level. Source: Researcher’s calculations, 2018 using MatlabR2017a Figure 4.4: Adjusted for thin trading returns, 5% VaR estimates and VaR exceedances estimates from 2007 to 2016 Similar diagrams for the various sub-periods; pre-holiday, post-holiday and normal are shown in Appendix C. The next section examined the VaR and CVaR for the various sub-periods of this research: pre- holiday, post-holiday and normal trading days. The VaR at 5%-quantile for each period were calculated and compared amongst one other. The results are displayed in Table 4.10: 80 University of Ghana http://ugspace.ug.edu.gh Table 4.10: 5% VaR and 5% CVaR estimates for thin trading adjusted returns for the period 03.01.2007 to 30.12.2016 under the various categories. 𝑃𝑟𝑒ℎ𝑜𝑙𝑖𝑑𝑎𝑦 𝑃𝑜𝑠𝑡ℎ𝑜𝑙𝑖𝑑𝑎𝑦 𝑁𝑜𝑟𝑚𝑎𝑙 𝑉𝑎𝑅0.05 0.0858 0.0827 0.5755 𝐶𝑉𝑎𝑅0.05 0.0878 0.0843 0.6220 Note: 𝐶𝑉𝑎𝑅0.05 = 𝑉𝑎𝑅0.05 + 𝑎𝑣𝑒𝑟𝑎𝑔𝑒(𝑒𝑥𝑐𝑒𝑒𝑑𝑎𝑛𝑐𝑒𝑠) Source: Researcher’s calculations, 2018. The results in Table 4.10 shows the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR) estimates at 5% significance level for the sub-periods considered under this study, they include the pre-holiday days, post-holiday days and the normal trading days. From the results, the risk measures for the normal trading days are higher than the other two sub-periods. Hence, in the worst 5% of returns, an investor’s average loss is approximately 8.43% during post- holiday days on the GSE and for an investor on the GSE, there is a 5% chance to lose about 8.78% of his or her return during pre-holiday days. This could be an indication that the high and abnormal returns observed in Table 4.10 above for the pre-holiday and post-holiday returns (in comparison to that of the normal trading days) could not be as a result of bearing higher risk as observed by Yuan and Gupta (2014) in China. However, it could be as a result of other unknown factors as iterated again by Yuan and Gupta (2014) as possible explanations for the other countries where their risk measures were relatively lower. 81 University of Ghana http://ugspace.ug.edu.gh Table 4.11: 5% VaR and 5% CVaR estimates for the adjusted for thin trading returns for the period 03.01.2007 to 30.12.2016 for the various holidays Variables NYPRE 0.01556 0.01556 NYPOST 0.00957 0.00997 INDEPRE 0.00375 0.00375 INDEPOST 0.00371 0.00371 EASTERPRE 0.00377 0.00417 EASTERPOST 0.00685 0.00685 WORKERPRE 0.00616 0.00657 WORKERPOST 0.00541 0.00541 AUPRE 0.00521 0.00521 AUPOST 0.00901 0.00901 REPUPRE 0.00834 0.00834 REPUPOST 0.00721 0.00721 FITRPRE 0.00792 0.00833 FITRPOST 0.00539 0.0062 ADHAPRE 0.00892 0.00892 ADHAPOST 0.00546 0.00546 FARMERPRE 0.00654 0.00694 FARMERPOST 0.01717 0.01757 XMASPRE 0.01797 0.01797 XMASPOST 0.01112 0.01112 FOUNDERPRE 0.00388 0.00428 FOUNDERPOST 0.00395 0.00395 NORMAL 0.5755 0.6219 Source: Researcher’s calculations, 2018 using MatlabR2017a. 82 University of Ghana http://ugspace.ug.edu.gh 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 VARIABLES VaR @ 5% CVaR @ 5% Source: Researcher, 2018. Figure 4.5: Bar chart of the 5% VaR and 5% CVaR estimates of the various holidays The Figure 4.5 above is based on Table 4.11 and from Figure 4.5, it is observed that the risk levels associated with some of the holidays (Workers post and Farmers pre) that had significant returns as observed in Table 4.8 are relatively lower. Workers post-holiday had a CVaR of approximately 0.541% and Farmers pre-holiday of about 0.694% are comparatively lower than that of the other holidays considered in this study. These are indications that the significant results shown on Table 4.8 for both Farmers pre-holidays and Workers day holiday may be as a result of other factors such as mood or closing effect. This notwithstanding, the Farmers post-holiday recorded the third highest CVaR of approximately 1.757% after that of the Christmas pre-holidays (1.797%) and Normal trading days (62.19%). This value is an indication that the abnormal high returns observed in Table 4.8 for Farmers post-holiday may be as a result of the risk of loss of returns. 83 FREQUENCY (%) NYPRE NYPOST INDEPRE INDEPOST EASTERPRE EASTERPOST WORKERPRE WORKERPOST AUPRE AUPOST REPUPRE REPUPOST FITRPRE FITRPOST ADHAPRE ADHAPOST FARMERPRE FARMERPOST XMASPRE XMASPOST FOUNDERPRE FOUNDERPOST University of Ghana http://ugspace.ug.edu.gh Generally, it is observed that all the holidays from both Table 4.11 and Figure 4.5 have VaR and CVaR values below 5%. Table 4.12: 5% VaR and 5% CVaR estimates for Ghana-specific and non-Ghana specific holidays 𝐺ℎ𝑎𝑛𝑎 − 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑁𝑜𝑛 − 𝐺ℎ𝑎𝑛𝑎 ℎ𝑜𝑙𝑖𝑑𝑎𝑦𝑠 − 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑜𝑙𝑖𝑑𝑎𝑦𝑠 𝑉𝑎𝑅0.05 0.0536 0.1148 𝐶𝑉𝑎𝑅0.05 0.0548 0.1173 Source: Researcher’s calculations, 2018 using MatlabR2017a. Table 4.13: 5% VaR and 5% CVaR estimates for pre- and post-holidays of Ghana- specific holidays and non-Ghana specific holidays Variables 𝑉𝑎𝑅0.05 𝐶𝑉𝑎𝑅0.05 GH. PRE 0.0223 0.0231 GH. POST 0.0318 0.0322 NON-GH. PRE 0.0606 0.0637 NON-GH. POST 0.0519 0.0532 Note: GH. –Ghana-specific, NON-GH- Non-Ghana specific. Source: Researcher’s calculations, 2018 using MatlabR2017a. From Tables 4.12 and 4.13, it is generally observed that the risk levels associated with the returns of Ghana- specific holidays are relatively lower than that of the non-Ghana specific holidays. This again, is an indication that the returns that contributed to the pre-holiday and post-holiday effects in Table 4.10 did not have their associated risk levels as a contributing factor. A robustness test on the standardized residuals ((𝑧𝑡) and the squared standardized residuals (𝑧2𝑡 )was performed and reported in Appendix D and shows there was no ARCH effect in the 84 University of Ghana http://ugspace.ug.edu.gh standardized residuals and squared standardized residuals. Whereas, there was no serial correlation in the squared standardized residuals, beyond the 7th lag there was evidence of relatively little serial correlation in the standardized residuals. The two variables showed that they were not normally distributed but leptokurtic. This is a confirmation that the ARMAX- GARCH model is an appropriate model for this study. 85 University of Ghana http://ugspace.ug.edu.gh CHAPTER FIVE SUMMARY, CONCLUSION AND RECOMMENDATIONS 5.1 Introduction This chapter presents the summary of the findings related to this study, conclusions, recommendations as well as potential areas for future research. The chapter is presented in five- fold sections beginning with this section which seeks to introduce the chapter. The next section gives a summary of how the research was conducted. Section 5.3 which is the conclusion gives a synthesis of the study. The last two sections are the recommendations and suggestions for future research. 5.2 Summary The primary aim of this study was to determine if there was a holiday effect (any or all of pre- holiday effects and post-holiday effects) in the Ghana Stock Exchange. The study considered the main financial data from the main index; GSE-All Shares Index (2007 to 2010) and the GSE-CI (2011 to 2016). The daily returns of the main index were computed from the daily closing value of the main index. The computed continuous compounded returns were further adjusted for thin trading by using the formulae proposed by Miller et al. (1994). This was necessary because thin trading was and is currently one of the major characteristics of emerging markets such as Ghana. The adjusted for thin trading returns were used for the analysis of the study. The diagnostic tests performed on the return distribution showed that the distribution was not normally distributed using the Jarque-Berra test, the residuals were serially correlated with the use of the Breusch-Pagan test. Using the White test, the variances of the error terms were heteroscedastic in nature and the returns were stationary with the use of the Augmented Dickey 86 University of Ghana http://ugspace.ug.edu.gh Fuller test. Also, these results were indications that the distribution of the financial data used was leptokurtic and asymmetric which are stylized characteristics of financial time series data. The results additionally, affirmed that the OLS may not be good model to fit this data and supported the use of the ARMAX-GARCH model, where the serial correlation were captured by the ARMAX model and the heteroscedasticity was removed by fitting a GARCH model. The study adopted the ARMAX (2, 2) - GARCH (1, 1) model with a 𝐺𝐿+ innovations to estimate the conditional mean and conditional variance equations. Finally, based on the residuals of the estimated parameters, the 5% VaR and 5% CVaR (ES) were calculated to determine the risk levels of each period and each holiday considered in the study. Despite its discovery in other countries such as the US, UK, China and South Africa, the holiday effect cannot be generalized to cover other markets including emerging ones. As a result, it was imperative to investigate the holiday effect on the Ghanaian stock exchange to ascertain if the market was efficient. 5.3 Conclusion There exist statistically significant positive pre-holiday effects and positive statistically significant post-holiday effects on the GSE. Furthermore, the study showed that the pre-holiday effects and post-holiday effects discovered occurred on the 7th trading day before and on the 7th trading day after a holiday. Significant pre- and post- holiday returns are implications that investors can take advantage by trading on the 7th trading day before and the 7th trading day after a holiday which means that there might be a possibility for investors to earn abnormal returns in these sub-periods. Again, whereas, the 7th trading day before and the 7th trading day after the Farmers day holiday contributed significantly to the pre-holiday and post-holiday effects respectively, only the 7th trading day after the International Workers day (Labour day) holiday contributed significantly 87 University of Ghana http://ugspace.ug.edu.gh to the post-holiday effects. This suggests that on these days, the average returns on the GSE are abnormally higher than the returns on normal trading days indicating that on these days’ investors tend to trade more. Closely related to the above, the study revealed that the Ghana-specific holidays which are defined as holidays that are celebrated uniquely and only recognized in Ghana have pre- and post-holiday effects. The non-Ghana specific holidays which are holidays that are celebrated both locally and internationally, on the other, recorded only a pre-holiday effect. The results show that generally only non-Ghanaian specific post-holiday returns are insignificant at 5%. The highest average return was documented in the Ghana-specific pre-holiday days, followed by non-Ghana specific pre-holiday and the Ghana-specific post-holiday had the least average return. This indicates that investors could take advantage of Ghana-specific pre-holiday days in terms of the holiday effect. Generally, investors are usually attracted to take on higher risks which usually come with higher returns. However, the results of the findings from this research show that, the significant abnormal returns observed in both the pre- and post-holidays sub-periods were not serving as compensation to existing investors on the Ghana Stock Exchange for taking higher level of risks because the risk measures for the normal trading days were higher than the other two sub- periods. Also, the Farmers post-holiday returns amongst the other holidays considered recorded the second highest Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) estimates, whereas the other 2 holiday days (Farmers pre-holiday and Workers post-holiday) that recorded significant abnormal returns; had relatively lower risks. These are indications that this could be the explanation for the abnormal returns observed during the Farmers post-holiday days and other factors such as mood, euphoria, closing effect could explain the Workers pre-holiday and Farmers pre-holiday abnormal returns. Again, the associated risk for non-Ghana specific pre- holiday days was the highest and the lowest was recorded for the Ghana-specific pre-holiday 88 University of Ghana http://ugspace.ug.edu.gh days. Hence, investors could take advantage of the Ghana-specific pre-holiday days because the risk associated with these average returns are minimal. Finally, the overall results showed that returns on the 7th day before International Workers day and the 7th day after either International Workers day or Farmers day or both were abnormally higher and statistically significant. These results indicate that on Workers day and Farmers day the GSE experienced stronger investor reactions than on the other holidays considered in this research, therefore, investors should trade on these days without necessarily worrying about their risk levels. Again, the results show that mood may not be the reason associated with the occurrence of the holiday effect since holidays such as Christmas, New Year and Easter had insignificant returns. 5.4 Recommendations With the discovery of the holiday effect on the GSE and an eventual indication of the inefficiency of the market, investors on the GSE can make abnormal returns by taking advantage of this calendar anomaly. With the indication of the inefficiency of the GSE, it is prudent for the regulatory bodies especially the Security and Exchange Commission (SEC) and the Ghana Stock Exchange (GSE) to formulate policies that are geared towards making the stock Exchange in Ghana efficient. These policies should encourage investors to take advantage of this anomaly because according to Malkiel (2003) a market can eventually become efficient if its inefficiencies are taking advantage of. 5.5 Study limitations and direction for further research The number of years considered for this study was 10, this period compared to similar works done by Ariel (1990), Kim and Park (1994) and Dodd and Gakhovich (2011) could be referred 89 University of Ghana http://ugspace.ug.edu.gh to as a short period. There is therefore the need for further studies to consider larger observations by using hourly returns. It will be intriguing to also investigate the significance of the holiday effect by controlling for other market anomalies such as the month-of-the-year and day-of-the week effects, this will determine if the holiday effect discovered is as result of other calendar anomalies. Additionally, investigating the holiday effect in relation to the following themes: firm size level, industry level, liquidity and its persistence over time will help determine which industry, firm size level experiences the holiday effect and if the holiday effect is persistent over time. Finally, investigating the spill over effects of public holidays in other nations such as Nigeria on the GSE will help know if holidays celebrated in other countries affect the way investors react on the GSE. 90 University of Ghana http://ugspace.ug.edu.gh REFERENCES Abd. Majid, M. S. (2017). Re-examination of calendar anomalies in the Indonesian stock market. Journal of Applied Economic Sciences, 11(8), 1714-1723. Alagidede, P. (2013). Month-of-the-year and pre-holiday effects in African stock markets. South African Journal of Economic and Management Sciences, 16(1), 64-74. Alagidede, P., & Panagiotidis, T. (2009). Calendar anomalies in the Ghana Stock Exchange. Journal of Emerging Market Finance, 8(1), pp. 1 - 23. Alkhazali, O. (2008). The impact of thin trading on day-of-the-week effect: Evidence from the United Arab Emirates. Review of Accounting and Finance, 7(3), 270-284. Alkhazali, O. (2011). Does infrequent trading make a difference on stock market efficiency? Evidence from the Gulf Cooperation Council. Studies in Economics and Finance, 28(2), 96-110. Al-Loughani, N., & Chappell , D. (2001). Modelling the day-of-the-week effect in the Kuwait Stock Exchange: A non-linear GARCH representation. Applied Financial Economics, 11, 353–59. Andoh, C. (2009). Stochastic variance models in discrete time with Feedforward Neural Networks. Neural Computations, 21, 1990-2008. Andoh, C. (2010). GARCH family models under varying innovations. Decision, 37(1), 22-55. Andoh, C., Mensah, L., & Atsu, F. (2018). GL+ and GL- Regressions. In: Anh L., Dong L., Kreinovich V., Thach N. (eds) Econometric for financial Applications. ECONVN 2018. Studies in Computational Intelligence, vol 760, pp. 63-77, Springer, Cham. Angabini, A., & Wasiuzzaman, S. (2011). GARCH models and the financia crisis: A study of the Malaysian stock market. The International Journal of Applied Economics and Finance, 5(3), pp. 226-236. Government of Ghana. (2011, June 24). Retrieved December 12, 2017, from Is Founder's Day worth celebrating?: http://www.ghana.gov.gh/index.php/media- center/features/1950-is-founder-s-day-worth-celebrating. Appiah-Kusi, J., & Menyah, K. (2003). Returns predictability in African stock markets. Review of Financial Economics, 12(3), 247-270. Ariel, R. (1990). High stock returns before holidays: Existence and evidence on possible causes. Journal of Finance, 45(5), 1611-1626. Ariss, T. R., Rezvanian, R., & Medhian, S. M. (2011). Are Gulf Cooperation Council (G.C.C.) stock markets special? Applied Financial Economics, 22(3), pp. 177-195. 91 University of Ghana http://ugspace.ug.edu.gh Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228. Africa Securities Exchange Association (ASEA). (2016). Annual Report, 2016, Nairobi, Kenya: ASEA. Balakrishnan, N. (1992). Handbook of the logistic distribution. Marcel Dekker, Inc., New York -Basel. Balakrishan, N., & Leung, M. Y. (1998). Order statistics from the Type 1 Generalized Logistic distribution. Commun. Statis. -Simula, 17,25-50. Beck, T., & Levine, R. (2004). Stock markets, banks and growth: Panel evidence. Journal of Banking and Finance, 28(3), 423-442. Bekaert, G., Erb, B. C., Harvey, R. C., & Viskanta, T. E. (1998). Distributional characteristics of emerging markes and asset allocation. Journal of Portfolio Management, 24(2), 102-116. Berkes, I., Horvath, L., & Kokoszka, P. S. (2003). GARCH processes: Structure and estimation. Bernoulli, 9, pp. 201-227 Berument, H., & Ceylan, N. B. (2012). Effects of football on stock markets: Return-volatility relationship. Social Science Journal, 49(3), 368-374. Berument, M. H., & Ceylan, N. (2013). Soccer and stock market risk: Empirical evidence from the Istanbul Stock Exchange. Psychological Reports, 112(3), 769-770. Boateng, Richard (2016). Research Made Easy. Classic Edition. Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics, 31, pp. 307-327. Brockman, P., & Michalyuk , D. (1998). The persistent holiday effect: Additional evidence. Applied Economics Letter, 5(2), 205-209. Brooks, C. (2008). Introductory Econometrics for Finance (2 ed.). Cambridge University Press, Cambridge, New York, NY. Campbell, J. Y., Lo, A. W., & Mackinlay, A. C. (1997). The Econometrics of Financial markets. (2 e.d.) Princeton: Princeton University Press. Chen, T.-C., & Chien, C.-C. (2011). Size effect in January and cultural influences in an emerging stock market: The perspective of behavioral finance. Pacific-Basin Finance Journal, 19(2), 208-229. Chien, C., Lee, C., & Wang, A. (2002). A note on stock market seasonality: The impact of stock price volatility on the application of dummy variable regression model. The Quarterly Review of Economics and Finance, Elsevier, 42(1), 155-162. 92 University of Ghana http://ugspace.ug.edu.gh Coutts, J., & Sheikh, M. (2002). The anomalies that aren't there: The weekend, January and pre-holiday effects on the All-Gold index on the Johannesburg Stock Exchange 1978- 1997. Applied Financial Economics, 12(12), 863-871. Danielson, J. (2011). Financial Risk Modelling. (1 ed.) . UK: John Wiley & Sons Ltd. Dickey, D. A. & Fuller, W. A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, pp. 427- 431. Dodd, O., & Gakhovich, A. (2011). The holiday effect in Central and Eastern European financial markets. Investment Management and Financial Innovations, 8(4), 1. Dorfman, M. S., & Cather, D. A. (2012). Introduction to risk management and insurance. (10 ed). Pearson Education. Dupernex, S. (2007). Why might share prices follow a random walk? Student Economic Review, 21. Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, pp. 987-1007. Esso, L. J. (2010). Re-examining the finance-growth nexus: Structural break, threshold cointegration and causality evidence from the ECOWAS. Journal of Economic Development, 35(3), 57-79. Fama, E. (1965). The behabiour of stock market prices. The Journal of Business, 38(1), 34- 105. Fama, E. (1970). Efficient capital markets: A review of theory and empirical work. Journal of Finance, 25(2), 383-417. Fields, M. J. (1931). Stock prices: A problem verification. The Journal of Business, 4, 415. Founders' day to be placed on Ghana's Holiday Calendar. (2012, September 22). Retrieved November 28, 2016, from Modern Ghana: https://www.modernghana.com/news/419363/1/founders-day-to-be-placed-on- ghanas-holiday-calend.html. Fountas, S., & Segredakis, K. (2002). Emerging stock market return seasonalities: The January effect and the tax-loss selling hypothesis. Applied Financial Economics, 12(4), 291-299. French, K. R. (1980). Stock returns and the weekend effect. Journal of Financial Economics, 8, 55-69. Frimpong, J. Magnus. (2008). Capital market efficiency: An analysis of weak-form efficiency on the Ghana Stock Exchange. Journal of Money, Investment and Banking (5). 93 University of Ghana http://ugspace.ug.edu.gh Frimpong, J. M., & Oteng-Abayie, E. F. (2006). Modelling and forecasting volatility of returns on the Ghana Stock Exchange Using GARCH models. American Journal of Applied Sciences, 3(10), 2042-2048. Gama, P., & Vieira, E. (2013). Another look at the holiday effect. Applied Financial Economics, 23(20), 1466-4305. Ghana Stock Exchange (GSE) Live- African Stock Exchange. Retrieved January 2, 2018, from https://afx.kwayisi.org/gsegh/. Ghana Stock Exchange (GSE). Retrieved January 2, 2018, from www.gse.com.gh. Ghana Stock Exchange (GSE), (2018). Annual Report 2018. Accra, Ghana. GSE Gibbons M. R., & Hess, P. (1981). Day-of-the-week effects and asset returns. The Journal of Finance, 54(4), 579-596. Greene, William H. (2003). Econometric Analysis (5ed). Prentice Hall. Göb, R. (2011). Estimating value-at-risk and conditional value-at-risk for count variables. Quality and Reliability Engineering International , 27(5), pp. 659-672. Gujarati, Domador, & Porter, D. (2009). Basic Econometrics. (2 ed). McGraw-Hill/ Irwin. Hayashi, Fumio. (2000). Econometrics. Princeton University Press, Princeton, N.J. Huang, J., Shieh, J., & Kao, Y.-C. (2016). Starting points for a new researcher in behavioral finance. International Journal of Managerial Finance, 21(1), 92-103. Hung, Jui-Cheng (2009). Deregulation and liberilazation of the Chinese stock market and the improvement of market efficiency. The Quarterly Review of Economics and Finance, 49(3), 843-857. Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression. (3 e.d.). Wiley & Sons. IRM. (2002). A Risk Management Standard. Retrieved from The Institute of Risk Management. Jacobs, B. I., & Levy, K. N. (1988). Calendar anomalies: Abnormal returns at calendar turning points. Financial Analyst Journal, 44(6), 28-39. Jackson, M. O., & Kremer, I. (2007). On the informational inefficiency of the discriminatory price auctions. Journal of Economic Theory, 132(1), 507-517. Jahfer, A. (2015). Calendar effects of Colombo markets. Journal of Management, 12(2), 121- 132. Jarrett, J. E. (2010). Efficient Market Hypothesis and daily variation in small Pacific-Basin stock markets. Management Research Review, 33(12), 1128-1139. 94 University of Ghana http://ugspace.ug.edu.gh Jefferis, K., & Smith, G. (2005). The changing efficiency of African stock markets. South African Journal of Economics, 73(1), 54-67. Jovanovic, F., Andreadakis, S., & Schinckus, C. (2016). Efficient market hypothesis and fraud on the market theory a new perspective for class actions. Research in International Business and Finance, 36, 177-190. Kargbo, S. M., & Adamu, P. A. (2009). Financial development and economic growth in Sierra Leone. Journal of Monetary and Economic Integration (9), 30-61. Keim, D. B. (1989). Trading patterns, bid-ask spreads and estimated security returns: The case of common stocks at calendar turning points. Journal of Financial Economics, 25,75-97. Kim, C., & Park, J. (1994). Holiday effects and stock returns: Further evidence. Journal of Financial and Quantitative Analysis, 29(1), 145-157. Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of unit root. Journal of Econometrics, 54, pp. 159-178. Kuttu, S. (2017). Modelling long memory in volatility in Sub-Saharan equity markets. Research in International Business and Finance, 44, 176-185. Lahav, Eyal, Shavit, Tal, & Benzion, U. (2016). Can't wait to celebrate: Holiday euphoria, impulsive behavior and time preference. Journal of Behavioral and Experimental Economics, 65, 128-134. Lakonishok, J., & Smidt, S. (1988). Are seasonal anomalies real? A ninety year perspective. Review of Financial Studies, 1(4), 403-425. Lee, Chien-Chiang, Lee, Jun-De & Lee, Chi-Chuan (2010). Stock prices and the efficient market hypothesis: Evidence from a panel stationary test with structural breaks. Japan and the World Economy, 22(1), 49-58. Liano, K., Marchand, P. H., & Huang, Gow-Cheng. (1992). The holiday effect in stock returns: Evidence from the OTC market. Review of Financial Economics, 2(1), pp. 45-54. Lim, Y. S., Ho, M. C., & Dollery, B. (2009). An empirical analyis of calendar anomalies in the Malaysian stock market. Applied Financial Economics, 20(3), 255-264. Loc, T., Lanjouw, G., & Lencsink, R. (2010). Stock market efficiency in thin-trading markets: The case of the Vietnamese stock Market. Applied Economics, 42(27), 3519- 3532. Magnusson , M., & Wydick , B. (2002). How efficient are Africa's emerging stock markets? Journal of Development Studies, 38(4), 141-156. 95 University of Ghana http://ugspace.ug.edu.gh Makridakis, S., & Hibon, M. (2000). The M3-Competition: results, conclusions and implications. International Journal of Forecasting, 16(4), 451-476. Malkiel, B. G. (2003). The Efficient Market Hypothesis and its critics. Journal of Economic Perspectives, 17(1), 59-82. Marrett, G. K., & Worthington, A. C. (2009). An empirical note on the holiday effect in the Australian Stock Market. Applied Economics Letters, 16(17), 1769-1772. Marshall, B., & Visaltanachoti , N. (2010). The other January Effect: Evidence against market efficiency. Journal of Banking and Finance, 34(10), 2413-2424. Masoud, N. M. (2013). The impact of stock market performance upon economic growth. International Journal of Economics and Financial Issues, 3(4), 788-798. McGowan, C., & Ibrihim, I. (2009). An analysis of the day-of-the-week in the Russia stock market. International Business & Economics Research Journal, 8(9), 25-30. Mehran, J., Meisami, A., & Busenbark, J. (2012). L'Chaim: Jewish holidays and stock market returns. Managerial Finance, 38(7), 641-652. Meneu, V., & Pardo, A. (2004). Pre-holiday effect, large trades and small investor behaviour. Journal of Empirical Finance, 11(2), 231-246. Mensah, L., Bokpin, G., & Owusu-Antwi, G. (2016). Time your investment on the Ghana Stock Exchange (GSE). African Journal of Economic and Management Studies, 7(2), 256-267. Miller, M., Muthuswamy, R., & Whaler, R. (1994). Mean reversion of Standard and Poor 500 Index basis changes: Arbitrage-induced or statistical illusion. The Journal of Finance, 49(2), 479-513. Mlambo, C., & Biekpe, N. (2007). The Efficient Market Hypothesis: Evidence from ten African stock markets. Investment Analysts Journal, 36 (66), 5-18. Nidhin, K., & Chandran, C. (2013). Importance of Generalized Logistic distribution in extreme value modeling. Applied Mathematics, 4, 560-573. Ntim, C. G., Opong, K. K., Danbolt, J., & Dewotor , F. S. (2011). Testing the weak‐form efficiency in African stock markets. Managerial Finance, Vol. 37 (3), 195-218. Obi, P., & Sil, S. (2013). VaR and time-varying volatility: A comparative study of three international portfolios. Managerial Finance, 39(7), pp. 625-640. Oh, H., & Lee, S. (2017). On change point test for ARMA-GARCH models: Bootsrap approach. Journal of the Korean Statistical Society. 96 University of Ghana http://ugspace.ug.edu.gh Olweny, T., & Kimani, D. (2011). Stock market performance and economic growth. Empirical evidence from Kenya using causality test approach. Advances in Management & Applied Economics, 1(3), 153-196. Pathirawasam, C., & Idirisinghe, I. (2011). Market efficiency, thin trading and non -linear behavior: Emerging market evidence from Sri Lanka. E a M Ekonomie a Management, 8(1), 112-122. Pearce, D. K. (1996). The robustness of calendar anomalies in daily stock returns. Journal of Economic and Finance, 20(3), 69-80. Pettengill, G. N. (1989). Holiday closings and security returns. Journal of Financial Research, 12(1), 57-67. Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75, pp. 335-346. Philpot, James, & Peterson, A. Craig. (2011). A brief history and recent developments in day- of-the-week effect literature. Managerial Finance, 37(9), pp. 808-816. Pickup, Mark. (2015). Quantitative Applications in the Social Sciences: Introduction to time series analysis. Thousand Oaks, CA: SAGE Publications Ltd. Pilbeam, K. (2010). Finance and Financial Markets. (2 ed.), UK: Palgrave Macmillan Ltd. Pisedasalasai, A., & Gunsekarage, A. (2007). Causal and dynamic relationships among stock returns, return volatility and trading volume: Evidence from emerging markets in South-East Asia. Asia-Pacific Financial Markets, 14, 277-297. Promislow, D. S. (2012). Fundamentals of actuarial mathematics. (2, Ed.) New Jersey, US: John Wiley & Sons Inc. Redja, G. E. (2014). Principles of risk management and insurance (12 ed.). Boston, U.S.: Pearson. Founders' day to be placed on Ghana's holiday calendar. (2012, September 22). Retrieved November 28, 2016, from Modern Ghana: https://www.modernghana.com/news/419363/1/founders-day-to-be-placed-on- ghanas-holiday-calend.html). Schindler, M. (2007). Rumors in financial markets: Insight into behavorial financial. Hoboken, NJ, USA: John Wiley & Sons, Inc. Schumpeter, J. A. (1911). The theory of ecoonomic development. Havard University Press, Cambridge. Security and Exchange Commission (SEC). (2016). Annual Report, 2016. Accra, Ghana: SEC. 97 University of Ghana http://ugspace.ug.edu.gh Shahid, Ali, & Akbar, M. (2009). Calendar effects in Pakistani stock market. Internatinal Review of Business Research Papers, 5(1), 389-404. Sharpe, W., Alexander, G., & Bailey, J. V. (1999). Investments (6 ed.). Englewood Cliffs, N.J.: Prentice Hall. Shefrin, H., & Statman, M., (2012). Behavioral finance in the financial crisis: Market efficiency, Minsky and Keynes. In Rethinking Finance: New Perspectives on the Crisis, Blinder, A. S., Lo, A. W., & Solow, R. M., New York , NY: Russell Sage Fooundation Publications. Sheppard, Kevin. MFE Toolbox. Retrieved from https://www.kevinsheppard.com/MFE_Toolbox on 9th March, 2018. Simons, Daniel, & Laryea, S. (2006). The efficiency of selected African stock markets. Finance India, 20(2), pp. 553-571. Statutory Public Holidays. (n.d.). Retrieved December 30, 2017, from Government of Ghana: http://www.ghana.gov.gh/index.php/news/52-map-of-natural-resources/306-statutory- public-holidays-2015). Taylor, S. (1986). Modelling Financial Time Series. Wiley, Chichester. Thaler, Richard H. (1987). Anomalies: The January effect. Journal of Economic Perspectives, 1(1), pp. 197-201. Tolikas, K. (2011). The rare event risk in African emerging stock markets. Managerial Finance, 37(3), 275-294. Tonchev, Dimitar, & Kim, Tae-Hwan. (2004). Calendar effects in Eastern European financial markets: Evidence from the Czech Republic, Slovakia and Slovenia. Applied Financial Economics, 14(14), 1035-1043. (n.d.). UNDP Annual Report 2013: Supporting Global Progress. UNDP. Vidanage, T., & Dayaratna-Banda, O. (2012). Does past information help predict future price movements in emerging capital markets? Evidence from the Colombo Securities Exchange. South Asia Economic Journal, Vol 13(Issue 2), pp. 241 - 264. Wasiuzzaman, S. (2017). Religious anomalies in Islamic stock markets: The Hajj effect in Saudi Arabia. Journal of Asset Management, 18(3), 157-162. Wasiuzzaman, S. (2018). Seasonality in the Saudi stock market: The Hajj effect. Quarterly Review of Economics and Finance, Elsevier, 67, 273-281. Wiredu, S., & Mamuna, I. (2015). Seasonal anomalies in stock returns in Ghana. Research Journal of Finance and Accounting, 6(14). World Bank, World Development Indicators (WDI). (2016). Listed domestic companies, 98 University of Ghana http://ugspace.ug.edu.gh Total [Data file]. Retrieved from https://data.worldbank.org/indicator/CM.MKT.LDOM.NO?view=chart. Accessed 27th February, 2019. Wright, William F., & Bower, Gordon. (1992). Mood effects on subjective probability assesssment. Organizational Behavior and Human Decision Process, 52, 276-291. Yuan, T., & Gupta, R. (2014). Chinese Lunar New Year effect in Asian stock markets, 1999- 2012. The Quarterly Review of Economics and Finance, Elsevier. 99 University of Ghana http://ugspace.ug.edu.gh APPENDICES APPENDIX A The Table a) below shows the OLS regression results based on the following regression model: ?̃?𝑡 = 𝛽0 + 𝛽1𝑃𝑟𝑒ℎ𝑜𝑙𝑖𝑑𝑎𝑦 + 𝛽2𝑃𝑜𝑠𝑡ℎ𝑜𝑙𝑖𝑑𝑎𝑦 + 𝛽3?̃?𝑡−1 + 𝜀𝑡 (A1) where: ?̃?𝑡 is the adjusted for thin trading return on day 𝑡; 𝛽𝑖 , 𝑖 = 0,1,2 are the average returns for normal trading days, the pre-holiday effects and post- holiday effects respectively, 𝛽3is the coefficient of the previous return term ?̃?𝑡−1 ; 𝑃𝑟𝑒ℎ𝑜𝑙𝑖𝑑𝑎𝑦 is a dummy variable which takes the value 1 for an 𝑛 number trading days before a holiday and 0 otherwise; 𝑃𝑜𝑠𝑡ℎ𝑜𝑙𝑖𝑑𝑎𝑦 is a dummy variable which takes the value 1 for an 𝑛 number trading days after a holiday and 0 otherwise and 𝜀𝑡 is the error term, 𝜀𝑡 ~ 𝑁(0, 𝜎 2). Table A1: OLS Regression Results DAY WINDOW VARIABLE ±1 ±2 ±3 ±4 ±5 ±6 ±7 ±8 -0.0036 -0.0095 0.0007 -0.0040 -0.0022 0.0047 -3.50E-05 0.0005 C (-0.2176) (-0.5858) (0.0455) (-0.2519) (-0.1373) (0.2950) (-0.0022) (0.0298) 0.9852*** 0.9828*** 1.0072*** 0.9999*** 0.9981*** 0.9780*** 0.9967*** 0.9786*** PRE (7.9020) (11.0866) (13.4549) (16.0003) (10.1344) (15.7102) (10.9655) (8.6465) 1.0020*** 0.9546*** 1.0432*** 1.0000*** 0.9990*** 0.8926*** 0.9907*** 0.9804*** POST (7.7780) (7.2590) (17.4392) (8.6465) (9.0626) (7.4750) (9.2846) (8.6527) ADJRETURNS -0.0144 -0.0111 0.0657*** 0.0019 -0.0070 -0.0325 -0.022 -0.0288 (-1) (-0.7313) (-0.5716) (3.4809) (0.1024) (-0.3604) (-1.7093) (-1.1379) (-1.519) R-Squared 0.0476 0.0683 0.1625 0.1182 0.0699 0.1141 0.0779 0.0588 AIC 2.4380 2.4159 2.3082 2.3609 2.4142 2.3656 2.4056 2.4262 HQC 2.4414 2.4194 2.3116 2.3643 2.4177 2.3690 2.4090 2.4296 SBIC 2.4474 2.4253 2.3176 2.3703 2.4236 2.3750 2.4150 2.4356 DW Stat. 2.0019 1.9876 1.9750 1.8505 1.9885 1.9803 2.0506 2.0157 Notes: T- Statistics are in parentheses. ***- 1% Significance level 100 University of Ghana http://ugspace.ug.edu.gh APPENDIX B STATIONARITY TEST RESULTS Table B1: ADF with intercept t-Statistic Prob.* Augmented Dickey-Fuller test statistic -12.38884 0.0000 Test critical values: 1% level -3.432807 5% level -2.862511 10% level -2.567332 *MacKinnon (1 996) one-sided p-values. Table B2: ADF results with intercept and trend t-Statistic Prob.* Augmented D ickey-Fuller test stati stic -12.4 1585 0.0 000 Test critical values: 1% level -3.961764 5% level -3.411629 10% level -3.127687 *MacKinnon ( 1996) one-sided p-va lues. Table B3: ADF results with neither an intercept nor a trend t-Statistic Prob.* Augmented Dickey-Fuller test statistic -12.39149 0.0000 Test critical values: 1% level -2.565900 5% level -1.940952 10% level -1.616613 *MacKinnon (1996) one-sided p-values. 101 University of Ghana http://ugspace.ug.edu.gh APPENDIX C The following figures depict the returns, VaR at 5% significance level and exceedances of the various periods considered in the study. Figure C1: Normal trading days 102 University of Ghana http://ugspace.ug.edu.gh Figure C2 Post-holiday day Figure C3: Pre-holiday days 103 University of Ghana http://ugspace.ug.edu.gh APPENDIX D: ROBUSTNESS TEST Autocorrelation Function (ACF) Partial Autocorrelation Function (PACF) Normality test Variables Stats. Prob.* 𝑧𝑡 0.1932 0.000 𝑧2𝑡 0.1247 0.000 104